# Tangent and normal lines to conics

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 4.B. Tangent and normal lines to conics Apollonius work on conics includes a study of tangent and normal lines to these curves. The purpose of this document is to relate his approaches to the modern viewpoints based upon analytic geometry and differential calculus. Classification of conics up to congruence Whenever one works with coordinate geometry, it is clearly useful to choose coordinate axes so that the algebraic equations defining the objects of study are as simple as possible. Geometrically, this essentially corresponds to the physical idea of moving the objects without changing their sizes or shapes. Linear algebra provides a powerful method for interpreting rigid motions in terms of coordinates. Specifically, if we define an abstract isometry of the coordinate plane R 2 as a transformation T of the form ) ( y1 y 2 = ( cos θ ε sin θ sin θ ε cos θ where θ is some real number (often we assume 0 θ 2π), ε = ±1, and (b 1, b 2 ) is a fixed vector in R 2, then two plane figures A and B are said to be congruent if there is an isometry T which maps A onto B. Additional comments on this concept appear on pages ( = document pages 10 17) of the following online document: ) ( x1 x 2 ) + ( b1 res/math133/geometrynotes2b.pdf We are particularly interested in finding a short list of conics so that every such curve defined by an equation of the form Ax Bx 1 x 2 + Cx Dx 1 + Ex 2 + F = 0 is congruent to one of the curves on the list. We need to impose some restrictions on the coefficients in order to eliminate unwanted examples; for example, if A = B = C = 0, then we have a first degree equation, and clearly we should find a way to eliminate such cases. If we do this, then basic results from a second linear algebra course imply that every curve defined by such an equation is congruent to an example with one of the following forms in which all coefficients except possibly F are nonzero. In each case, we might as well assume that A is positive. Ax F = 0 Ax Cx F = 0 Ax Ex 2 = 0 This is just a notational reformulation of a result (the Rigid Change of Coordinates Theorem) on page of the following document: res/math132/linalgnotes.pdf As indicated in the table on page 82 of the same document, the first equation defines either a line or a pair of lines, and it is of no interest to us here. The third equation always defines a parabola. However, the solutions to an equation of the second type depend upon the signs of the various coefficients: 1 b 2 )

2 (a) If C is positive and F 0, then the equation either defines a single point when F = 0 (the origin) or no points at all when F < 0, so this case is also of no interest to us. (b) If C is positive and F > 0, then the equation defines and ellipse when C A and a circle when C = A. (c) If C is negative and F 0, then the equation defines a hyperbola. (d) If C is negative and F = 0, then the equation defines a pair of lines which intersect at the origin, so this case is also of no interest to us. Thus we see that every conic of interest to us is congruent to either a parabola, an ellipse (or circle), or a hyperbola given by a fairly standard equation, and as usual we may rewrite these equations as follows: (ELLIPSE) x 2 + y2 b 2 = 1 (a, b > 0) (HYPERBOLA) x 2 y2 b 2 = 1 (a, b > 0) (PARABOLA) y = ax 2 (a 0) Parametrizations of conics The first point to note is that ellipses, hyperbolas and parabolas can all be represented by parametrized curves of the form s(t) = ( x(t), y(t) ) where x(t) and y(t) are differentiable functions; strictly speaking, a hyperbola is given by two parametrized curves corresponding to its two branches, and we shall have to be a bit careful about this as we proceed. In any case, the tangent line to the curve at the point s(t 0 ) is represented parametrically by s(t 0 ) + u s (t 0 ), where s (t 0 ) is given by taking the derivatives of the coordinate functions where x(t) and y(t). This parametric representation will define a line if s (t 0 ) (0, 0), and we claim this is true for suitably defined parametrizations of ellipses, hyperbolas and parabolas. Examples. 1. For the ellipse given by x 2 + y2 b 2 = 1 (a, b > 0) one has the parametrization (a cos t, b sin t), and for each t 0 we have s (t 0 ) = ( a sin t 0, b cos t 0 ). The right hand expression is never the zero vector because the sine and cosine functions are never simultaneously zero. 2. If we now consider the hyperbola given by x 2 y2 b 2 = 1 (a, b > 0) one has the parametrization (± a cosh t, b sinh t) in terms of the hyperbolic functions sinh u and cosh u; as noted before, this curve has two branches, and the ± signs correspond to parametrizations of the pieces of the curve in the half-planes x 0 and x 0. For each t 0 we have s (t 0 ) = 2

3 (± a sinh t 0, b cosh t 0 ). The right hand expression is never the zero vector because the hyperbolic cosine function is never zero (recall that cosh 2 u sinh 2 u = 1). 3. Finally, for the the parabola y = ax 2 (where a 0) we have the graph parametrization s(t) = (t, at 2 ); in this case s (t 0 ) = (1, 2at 0 ) and the latter is never zero because its first coordinate is always nonzero. Tangent lines to conics Apollonius approach to tangent lines for conics is typical of Greek geometry, and it is based upon the fact that such lines meet the curves only at the point of contact. This viewpoint is evident in elementary geometry, where one views a tangent line to a circle as a line which meets the circle in exactly one point. Experimetation with ellipses, parabolas and hyperbolas suggests that the same is true for these curves. The purpose of this section is to prove this property using the definition of tangent lines from calculus. Our discussion of tangent lines will require the concept of the two open half-planes determined by the line (less formally, the two sides of the line). Specifically, if the line is given by an equation of the form f(x, y) = Lx + My + n = 0 where (L, M) (0, 0), then two points (u 0, v 0 ) and (u 1, v 1 ) not on the line are said to lie on the same half-plane (or side or the line) provided f(u 0, v 0 ) and f(u 1, v 1 ) are either both positive or both negative; the points are said to lie on opposite half-planes or sides if one is positive and the other is negative. For example, the two sides of the x-axis are defined by the inequalities y > 0 and y < 0. In our discussion below, we shall need to know that an isometry has the following two properties: (1) If T is an isometry and L is a line which determines half-planes H + and H, then the image of L under T is a line, say M, and T maps H + and H onto the two half-planes determined by M. (2) If T is an isometry and s(t) is a continuous parametrized curve such that s (t 0 ) is defined, then the composite T o s(t) also has these properties, and in fact (T o s) (t 0 ) is equal to T ( s (t 0 ) ). Both proofs are fairly straighforward and left to the reader. Here is the result implying that all for each point p of a conic curve, all the remaining points lie on one side of the tangent line at p; strictly speaking, one must modify this slightly in the hyperbolic case so that the conclusion only implies to the unique branch of the hyperbola on which p lies. THEOREM. Suppose that Γ is either a parabola, a circle, an ellipse, or a branch of a hyperbola. Let p be a point of Γ, and let L be the tangent line to Γ at p. Then all points of Γ except p lie on ONE of the half-planes determined by L. Proof. The argument splits into cases depending upon the type of the conic. Since tangents and half-planes are preserved under isometries, it is enough to prove the results for the standard examples of conics described above. Suppose first that Γ is a parabola; we shall assume that in the equation y = ax 2 the coefficient a is positive; the other case will follow by the change of variables u = x, v = y. Let (c, ac 2 ) be a point on the parabola, so that the tangent line is defined by the equation y ac 2 = 2ac(x c). We claim that all other points on the parabola belong to the half-plane defined by the inequality 3

4 y ac 2 > 2ac(x c). Let t be an arbitrary real parameter value; then the question is whether at 2 ac 2 is greater than 2ac(t c) for t c. But (at 2 ac 2 ) 2ac(t c) = at 2 2act + ac 2 = a(t c) 2 and the latter is clearly positive if t c. Now suppose that Γ is a circle or an ellipse with the parametrization (a cos t, b sin t). We shall use the following result to describe the tangent line to the ellipse at a typical point: If we are given a curve which satisfies an equation f(u, v) = 0 where f(u 0, v 0 ) = 0 but f(u 0, v 0 ) 0, then the tangent line to the curve at p = (u 0, v 0 ) is defined by the vector equation f(p) (x p) = 0, and the normal direction at p is determined by f(p). This can be found in nearly any calculus book which covers partial derivatives. Applying the preceding result to the ellipse, we see that the normal direction to the ellipse at a point (x 0, y 0 ) is given by (2x/, 2y/b 2 ), and therefore the tangent line at the point p = (a cos θ, b sin θ) is defined by the following equation: ( ) ( ) 2a cos θ 2b sin θ (u a cos θ) + (v b sin θ) = 0 We claim that the expression on the right hand side is negative for all points (u, v) on the ellipse except for p. But if we make the substitution (u, v) = (a cos t, b sin t) in the expression on the left hand side of the equation and simplify by cancellations of the form c 2 /c 2 = 1, the left hand side reduces to (cos θ cos t cos 2 θ) + (sin θ sin t sin 2 θ) = cos(t θ) 1. This expression is always nonnegative, and it is zero if and only if t is an integral multiple of 2π; since (u, v) = p if t is an integral of 2π, it follows that all other points of the ellipse lie on the same side of the tangent line at p. A similar analysis applies to the branches of the hyperbola. Since the change of variables u = x, v = y interchanges the two branches, by symmetry it will suffice to consider the right hand branch on which x > 0 and the branch is parametrized by the curve (a cosh t, b sinh t). Suppose that p has coordinates (a cosh t 0, b sinh t 0 ). Then as in the elliptical case the normal direction to the hyperbola branch at a point (x 0, y 0 ) is given by the gradient expression (2x/, 2y/b 2 ), and therefore the tangent line at the point p = (a cosh t 0, b sinh t 0 ) is defined by the following equation: ( ) ( ) 2a cosh t0 2b sinh t0 (u a cosh t 0 ) (v b sinh t 0 ) = 0 If we now make the substitution (u, v) = (a cosh t, b sinh t) and simplify once again, we see that the expression on the left side is equal to (cosh t 0 cosh t cosh 2 t 0 ) (sinh t 0 sinh t sinh 2 t 0 ) = cosh(t θ) 1. Since cosh w 1 with equality if and only if w = 0, it follows as before that all points on the branch of the hyperbola except for p must lie on the same side of the tangent line at p. b 2 b 2 4

5 Differential calculus and normal lines to curves By definition, a normal line to a curve at a given point p on the curve is a line through p which is perpendicular to the tangent line at p. Apollonius work on conics includes a study of normal lines to these curves; it is based upon the observation that the shortest or longest distance from an external point to a curve is a given by a line segment which is perpendicular to the tangent line to the conic at the point of contact. For example, if the curve is a circle, then the line segment lies in the line joining the external point to the center of the circle; by elementary geometry, we know that this line is perpendicular to the tangent line at the point where the lines meet the circle. Our goal here is to give a simple proof of the general fact using calculus and parametric equations and vector representations for curves. The relationship between normal lines and maximum or minimum distances is then given by the following result: PROPOSITION. Suppose that s(t) = ( x(t), y(t) ) is a parametrized curve such that the coordinate functions x(t), y(t) are differentiable everywhere and for each value of t at least one of x (t), y (t) is nonzero. Let p = (c, d) be a point not on the curve, and suppose that t 0 is such that the distance from s(t 0 ) to p is minimized or maximized. Then the direction of the line joining these two points is perpendicular to s (t 0 ). Proof. We shall need the following vector differentiation identity from first year calculus: (a b) = a b + a b Since the distance between two points is nonnegative, the problem of maximizing or minimizing the distance is equivalent to maximizing or minimizing the square of the distance, and using vectors we can write the square of the distance between p and a point s(t) on the curve as δ 2 (t) = (s(t) p) (s(t) p). We can now use differential calculus to search for a minimum by setting δ 2(t) = 0, and if we apply the previously stated differentiation identity to the expression for δ 2 we obtain 0 = δ 2 (t) = 2 s (t) (s(t) p). Since s(t) p is the direction of the line joining s(t) and p, this is exactly the conclusion that we want. The proof of the proposition has the following simple implication. PROPOSITION. Suppose that s(t) = ( x(t), y(t) ) is a parametrized curve such that the coordinate functions x(t), y(t) are differentiable everywhere and for each value of t at least one of x (t), y (t) is nonzero. Let p = (c, d) be a point not on the curve, and let δ 2 (t) denote the square of the distance from p to s(t). Then t 0 is a critical point of δ 2 if andn only if the direction of the line joining p to s(t 0 ) is perpendicular to s (t 0 ). This is an immediate consequence of the formula for δ 2(t). Critical points of squared distance functions Let Γ be a nonsingular conic, and let p be a point in the plane. The first goal is to show that the squared distance function δ 2 always has an absolute minimum and therefore always has at least one critical point. 5

6 The case of an ellipse. In this case the distance squared function δ 2 satisfies the periodicity identity δ 2 (t + 2π) = δ 2 (t), so over the interval [0, 2π] it has both maximum and minimum values. Since δ 2 is periodic, these are maximum and minimum values for the function over the entire real line. The case of a parabola. In this case δ 2 (t) is a fourth degree polynomial whose highest degree term has the form a 4 t 4 where a 0. Every such polynomial function has an absolute minimum value over the entire real line; in particular, we have lim t ± δ 2 (t) = + (the same is true for every even degree polynomial function for which the coefficient of the highest degree term is positive). The case of a branch of a hyperbola. We shall only consider the branch of the standard hyperbola which lies in the half-plane x > 0; symmetry conditions will imply the same sorts of conclusions for the other branch. If p = (c, d), then δ 2 (t) = (cosh t c) 2 + (sinh t d) 2 (cosh t c) 2, and since the right hand side goes to infinity as t ±, we can argue as in the parabolic case that δ 2 has an absolute minimum over the real line: Specifically, let v be some value that δ 2 takes, and using the limit property choose M > 0 so large that δ 2 (t) v + 1 for t M. Then δ 2 takes an absolute minimum value on the interval [ M, M], and by choice of M this value is taken at an interior point of the interval. Since this minimum value is no greater than v and δ 2 (t) > v if t M, this minimum must be the minimum value of the function over the entire real line. Counting normals from a point to a conic. Having shown that there is always at least one normal to a conic through a given point in the plane, it is natural to ask two further questions: (1) Given a point p and a conic Γ, how many normals to Γ pass through the point p? (2) Given a point p on the conic Γ and a normal line L to Γ through p, are there other points q on L Γ such that L is also a normal to Γ at q? There are simple answers to both questions if the conic is a circle. PROPOSITION. Let Γ be a circle, and let p be a point in the coordinate plane. (i) If p is the center of Γ, then every line through p is a normal to Γ. (ii) If p is not the center of Γ, then there is exactly one normal line to Γ through p, and it contains the center of the circle. In either case, if L is a line through p which is normal to Γ, then L meets Γ in a second point q and is also a normal to Γ at q. Proof. The final statement follows from the preceding two because every the latter show that every normal line through Γ passes through the center of the circle, so that every such line meets the circle at two points and is normal to the circle at each of these points. Statement (i) follows because tangent lines to a circle are perpendicular to radial lines at their points of contact. To prove (ii), we first use an isometry to find a congruent circle centered at the origin and note that it will suffice to prove the result for the resulting circle, which has an equation of the form x 2 + y 2 = r 2 for some r > 0. We are assuming that p is not the center, so we may set p equal to (b cos α, b sin α) where b > 0 and α is some real number. There are two cases, depending upon whether or not p lies on the given circle. If p lies on the circle, then the radial line through p is the unique line through p which is perpendicular to the tangent line at p, and since it passes through the center of the circle the conclusion of (ii) is true in this case. 6

7 Suppose now that p does not lie on the given circle, so that b r. For each point (r cos θ, r sin θ) on the circle, the direction of the normal line is determined by the vector (sin θ, cos θ), and therefore the line joining p to (r cos θ, r sin θ) is perpendicular to the circle at the latter point if and only if 0 = (sin θ, cos θ) (b cos α r cos θ, b sin α r sin θ). Since (cos θ, sin θ) and (sin θ, cos θ) are perpendicular, this equation reduces to 0 = b cos α sin θ b sin α cos θ = b sin(θ α) and since b 0 the latter reduces to sin(θ α) = 0, which is true if and only if θ α is an integral multiple of π. Since sin(α + π) = sin α and cos(α + π) = cos α, it follows that a normal line to the circle through p = (b cos α, b sin α) meets the circle at either (r cos α, r sin α) or ( r cos α, r sin α). But the line through p and one of these points also goes through the other and through the center of the circle, which implies the conclusion of (ii). For the other conics, the answer to the first question is more complicated. If the conic Γ is not a circle, then there are always at most four normals to the conic from a given point p, but the exact number of normals varies in each case and depends upon the position of p. In one of the exercises for this unit, this question is studied when Γ is the standard parabola y = x 2 and p lies on the y-axis; for some choices of p there are three normals to the parabola, but for others there is only one. Here are some additional references: Finally, we address the question of whether a single line can be a normal line to several different points on a conic which is not a circle. The next result implies that a line can be a normal to at most two points of a conic. PROPOSITION. Let Γ be a nonsingular conic, and let L be a line. Then L and Γ have at most two points in common. As in other cases, it suffices to verify this result when Γ is one of the standard examples. If L is given by the algebraic equation px + qy = k where (p, q) (0, 0) this amounts to showing that for each of the quadratic equations in two variables defining the standard examples y = ax 2, b 2 x 2 + y 2 = b 2, b 2 x 2 y 2 = b 2 there are at most two simultaneous solutions of the system determined by the quadratic equation and the equation of the line. We can solve px + qy = k for x in terms of y or vice versa; it will be convenient to call the solved-for variable the constrained variable and the other variable the free variable. Substitution of for the constrained variable in terms of the free variable yields a quadratic equation in the free variable, which has at most two solutions. For each of these solutions there is a unique corresponding value of the constrained variable, and thus there are at most two simultaneous solutions for the given system(s) of equations. Here is the result on lines which are normal at two points of a conic. 7

8 PROPOSITION. Let Γ be a conic which is not a circle, and let L be a line. Then L is a normal line to Γ at two distinct points if and only if Γ is an ellipse or hyperbola and L is one of its axes. In particular, there are two such lines for an ellipse (which is not a circle) and one for a hyperbola, but none for a parabola. Proof. The argument breaks down into cases depending upon whether Γ, is an ellipse, a hperbola, or a parabola. In each case we shall look for conditions under which two points on the curve have the same normal direction (up to sign as usual), and in each case where this happens we shall show that the normal lines at the two points must be different. The case of a parabola. The curve s equation is y = ax 2 where a > 0. We shall show that no two points on the curve have the same normal direction. The tangent vector at a typical point (t, at 2 ) has the form (1, 2at), so the normal direction at that point is given by (2at, 1). If (s, as 2 ) is a second point on the curve (so s t), then (t, at 2 ) and (s, as 2 ) will have the same normal direction if and only if (2at, 1) and (2as, 1) are nonzero scalar multiples of each other. If c is a scalar such that the first vector is c times the second, then we have (2at, 1) = (2cas, c) and by equating coefficients we get c = 1. But this means that 2as = 2at, which implies s = t, contradicting our choice of s t. Therefore different points on the parabola have different normal directions, so no line can be normal to the parabola at two distinct points. The case of an ellipse. The curve is given in parametric form by (a cos t, b sin t) where a, b > 0 and a b since we are excluding the case of a circle. In this case the normal direction is determined by (b cos t, a sin t). If t is an integral multiple of 1 2π, then the normal line is one of the coordinate axes, so that the point is a vertex of the ellipse and the normal goes through the opposite vertex, so we knows what happens in such cases and henceforth we shall only consider cases where t is not an integral multiple of 1 2π. This implies that both cos t and sin t are nonzero. If u is a now point such that (b cos u, a sin u) = c(b cos t, a sin t) for some nonzero scalar k, then it follows that cos u = k cos t and sin u = k sin t; using the identity sin 2 + cos 2 = 1 we can conclude that 1 = k 2, so that the only nontrivial possibility is k = 1, and in fact this is realized when u = t + π. To conclude the proof in this case, it is enough to show that the normal line to the ellipse at (a cos u, b sin u) = ( a cos t, b sin t). If it did, then there would be a scalar s such that (a cos t, b sin t) + s (b cos t, a sin t) = ( a cos t, b sin t) so that a + sb = a and b + sa = b; in deriving the last two equations we use the fact that both sin t and cos t are nonzero. If we solve the pair of equations we obtain s = 2a b = 2b a and if we multiply both sides of the last equation by ab/2 we find that = b 2. Since we specifically assumed that b 2, this means that there is no value of s which solves the equation (a cos t, b sin t) + s (b cos t, a sin t) = ( a cos t, b sin t) and therefore the normal lines to the curve at parameter values t and t + π have no points in common. Therefore, if a normal line to an ellipse is not an axis, then it is not normal to the ellipse at any other point on the curve. The case of a hyperbola. This is similar to the preceding case, but now the curve s two branches are given in parametric form by (εa cosh t, b sinh t) where a, b > 0 and ε = ± 1. In this case 8

9 the direction of the normal line is determined by (εb cosh t, a sinh t), and once again the question is to see if there is a second point at which the normal direction is the same; we shall write this point as (ηb cosh u, a sinh u), where η = ± 1 does not depend upon ε. If k is the proportionality constant, then as before we must have ε cosh t = kη cosh u, sinh t = k sinh u. As in the elliptical case, it is useful to dispose of cases where the points lie on the x-axis (note that no points lie on the y-axis). We know that the x-axis meets the hyperbola normally at the two vertex points (±a, 0), so this case is understood. Therefore we shall assume that sinh t > 0 for the rest of the discussion; since cosh 2 sinh 2 = 1 and cosh 1, this also yields cosh t > 1. Also as in the elliptical case, given a point (εa cosh t, b sinh t) with t 0, it follows that the same normal direction arises for a second point on the hyperbola of the form ( εa cosh t, b sinh t) = (εa cosh t, b sinh t) since cosh t is even and sinh t is odd; note that this point lies on the other branch of the hyperbola. We would like to imitate the preceding argument and show that (i) the same normal direction does not arise for any other point on either branch of the hyperbola, (ii) the normal to the hyperbola at (εa cosh t, b sinh t) does not contain the point (εa cosh t, b sinh t). The sign factors ε and η introduce complications not present in the elliptical case, but we can reduce everything to the case where ε = 1 because the hyperbola is symmetric with respect to the isometry (reflection about the y-axis) sending (x, y) to ( x, y). Thus we shall also assume that ε = +1 for the remainder of this argument. To dispose of (i), we need to show that if k, η and u are such that (b cosh t, a sinh t) = (kηb cosh u, ka sinh u) then k = ±1, u = kt and η = k. Set p = cosh t and q = sinh t, so that p > 1 and q > 0, with p 2 = 1 + q 2. Since a and b are positive, equating the coefficients on both sides of the displayed equation yields p = kη cosh u and q = k sinh u, so that p 2 = k 2 cosh 2 u = k 2 + k 2 sinh 2 u = k 2 + q 2 and if we subtract p 2 = 1 + q 2 from this we get k 2 1 = 0, or k = ±1. Since sinh ku = k sinh u for k = ±1 and sinh is strictly increasing, if k = 1 then the coordinate equation a sinh t = ka sinh u = a sinh u implies that u = t (recall that we have reduced things to cases where sinh t, sinh u 0), and b cosh t = kηb cosh t = kηb cosh t = ηb cosh t implies η = 1 = k (since cosh t, cosh u > 1). Now suppose that k = 1. Then the coordinate equations yield a sinh t = a sinh u, so that t = u, and b cosh t = ηb cosh t, so that η = 1 = k. This completes the verification of (i). The normal line at (a cosh t, b sinh t) is given by the parametric form (a cosh t, b sinh t) + s (b cosh t, a sinh t) and the verification of (ii) amounts to saying there is no choice of s for which this expression equals ( a cosh t, b sinh t). This argument is parallel to the elliptical case. If we can find a scalar s so that (a cosh t, b sinh t) + s (b cosh t, a sinh t) = ( a cosh t, b sinh t) 9

10 so that a + sb = a and b sa = b; in deriving the last two equations we use the fact that both sinh t and cosh t are nonzero. If we solve the pair of equations we obtain s = 2a b = 2b a and if we multiply both sides of the last equation by ab/2 we find that = b 2. Since a, b > 0 this is impossible, and hence there is no value of s which solves the equation (a cosh t, b sinh t) + s (b cosh t, a sinh t) = ( a cosh t, b sinh t) and therefore the normal to the hyperbola at the point (εa cosh t, b sinh t) does not contain the point (εa cosh t, b sinh t). To sum up the entire discussion for hyperbolas, if a normal line to a hyperbola is not an axis, then it is not normal to the hyperbola at any other point on the curve. Addendum: Intersections of two conics In history04y.pdf we mentioned that two conics have at most four points in common. For a given pair of examples, it is usually fairly straightforward to do this using standard algebraic techniques from precalculus and earlier courses, but for the sake of completeness we note that a very abstract and general result of this type appears in Exercise 11 on pages (= document pages 33 34) of the following online document: res/progeom/pgnotes07.pdf 10

### Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

### Roots and Coefficients of a Quadratic Equation Summary

Roots and Coefficients of a Quadratic Equation Summary For a quadratic equation with roots α and β: Sum of roots = α + β = and Product of roots = αβ = Symmetrical functions of α and β include: x = and

### MATHEMATICS (CLASSES XI XII)

MATHEMATICS (CLASSES XI XII) General Guidelines (i) All concepts/identities must be illustrated by situational examples. (ii) The language of word problems must be clear, simple and unambiguous. (iii)

### Section 1.1. Introduction to R n

The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to

PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

### 2. THE x-y PLANE 7 C7

2. THE x-y PLANE 2.1. The Real Line When we plot quantities on a graph we can plot not only integer values like 1, 2 and 3 but also fractions, like 3½ or 4¾. In fact we can, in principle, plot any real

### Linear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices

MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two

### Understanding Basic Calculus

Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other

### Copyrighted Material. Chapter 1 DEGREE OF A CURVE

Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two

### South Carolina College- and Career-Ready (SCCCR) Pre-Calculus

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know

### x 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1

Implicit Functions Defining Implicit Functions Up until now in this course, we have only talked about functions, which assign to every real number x in their domain exactly one real number f(x). The graphs

### Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.

Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.

### 1.7 Graphs of Functions

64 Relations and Functions 1.7 Graphs of Functions In Section 1.4 we defined a function as a special type of relation; one in which each x-coordinate was matched with only one y-coordinate. We spent most

### Section 10.7 Parametric Equations

299 Section 10.7 Parametric Equations Objective 1: Defining and Graphing Parametric Equations. Recall when we defined the x- (rcos(θ), rsin(θ)) and y-coordinates on a circle of radius r as a function of

### Algebra 1 Course Title

Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept

### Functions and Equations

Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c

### 2 Complex Functions and the Cauchy-Riemann Equations

2 Complex Functions and the Cauchy-Riemann Equations 2.1 Complex functions In one-variable calculus, we study functions f(x) of a real variable x. Likewise, in complex analysis, we study functions f(z)

Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function

### TOPIC 3: CONTINUITY OF FUNCTIONS

TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let

### Review of Fundamental Mathematics

Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools

### 3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes

Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general

### GRAPHING IN POLAR COORDINATES SYMMETRY

GRAPHING IN POLAR COORDINATES SYMMETRY Recall from Algebra and Calculus I that the concept of symmetry was discussed using Cartesian equations. Also remember that there are three types of symmetry - y-axis,

### Geometric Transformations

Geometric Transformations Definitions Def: f is a mapping (function) of a set A into a set B if for every element a of A there exists a unique element b of B that is paired with a; this pairing is denoted

### Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson

### Math 241, Exam 1 Information.

Math 241, Exam 1 Information. 9/24/12, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.1-14.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)

### String Art Mathematics: An Introduction to Geometry Expressions and Math Illustrations Stephen Arnold Compass Learning Technologies

String Art Mathematics: An Introduction to Geometry Expressions and Math Illustrations Stephen Arnold Compass Learning Technologies Introduction How do you create string art on a computer? In this lesson

### (a) We have x = 3 + 2t, y = 2 t, z = 6 so solving for t we get the symmetric equations. x 3 2. = 2 y, z = 6. t 2 2t + 1 = 0,

Name: Solutions to Practice Final. Consider the line r(t) = 3 + t, t, 6. (a) Find symmetric equations for this line. (b) Find the point where the first line r(t) intersects the surface z = x + y. (a) We

### The Method of Partial Fractions Math 121 Calculus II Spring 2015

Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method

### We call this set an n-dimensional parallelogram (with one vertex 0). We also refer to the vectors x 1,..., x n as the edges of P.

Volumes of parallelograms 1 Chapter 8 Volumes of parallelograms In the present short chapter we are going to discuss the elementary geometrical objects which we call parallelograms. These are going to

### 1 if 1 x 0 1 if 0 x 1

Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

### Student Performance Q&A:

Student Performance Q&A: AP Calculus AB and Calculus BC Free-Response Questions The following comments on the free-response questions for AP Calculus AB and Calculus BC were written by the Chief Reader,

### Solutions to old Exam 1 problems

Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections

### WASSCE / WAEC ELECTIVE / FURTHER MATHEMATICS SYLLABUS

Visit this link to read the introductory text for this syllabus. 1. Circular Measure Lengths of Arcs of circles and Radians Perimeters of Sectors and Segments measure in radians 2. Trigonometry (i) Sine,

### 4B. Line Integrals in the Plane

4. Line Integrals in the Plane 4A. Plane Vector Fields 4A-1 Describe geometrically how the vector fields determined by each of the following vector functions looks. Tell for each what the largest region

### Determine whether the following lines intersect, are parallel, or skew. L 1 : x = 6t y = 1 + 9t z = 3t. x = 1 + 2s y = 4 3s z = s

Homework Solutions 5/20 10.5.17 Determine whether the following lines intersect, are parallel, or skew. L 1 : L 2 : x = 6t y = 1 + 9t z = 3t x = 1 + 2s y = 4 3s z = s A vector parallel to L 1 is 6, 9,

### Multiplicity. Chapter 6

Chapter 6 Multiplicity The fundamental theorem of algebra says that any polynomial of degree n 0 has exactly n roots in the complex numbers if we count with multiplicity. The zeros of a polynomial are

### Section 13.5 Equations of Lines and Planes

Section 13.5 Equations of Lines and Planes Generalizing Linear Equations One of the main aspects of single variable calculus was approximating graphs of functions by lines - specifically, tangent lines.

### 14. GEOMETRY AND COORDINATES

14. GEOMETRY AND COORDINATES We define. Given we say that the x-coordinate is while the y-coordinate is. We can view the coordinates as mappings from to : Coordinates take in a point in the plane and output

### Solutions to Homework 10

Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x

### L 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has

The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:

### Vocabulary Words and Definitions for Algebra

Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

### Guide to SRW Section 1.7: Solving inequalities

Guide to SRW Section 1.7: Solving inequalities When you solve the equation x 2 = 9, the answer is written as two very simple equations: x = 3 (or) x = 3 The diagram of the solution is -6-5 -4-3 -2-1 0

### MATH SOLUTIONS TO PRACTICE FINAL EXAM. (x 2)(x + 2) (x 2)(x 3) = x + 2. x 2 x 2 5x + 6 = = 4.

MATH 55 SOLUTIONS TO PRACTICE FINAL EXAM x 2 4.Compute x 2 x 2 5x + 6. When x 2, So x 2 4 x 2 5x + 6 = (x 2)(x + 2) (x 2)(x 3) = x + 2 x 3. x 2 4 x 2 x 2 5x + 6 = 2 + 2 2 3 = 4. x 2 9 2. Compute x + sin

### JUST THE MATHS UNIT NUMBER 8.5. VECTORS 5 (Vector equations of straight lines) A.J.Hobson

JUST THE MATHS UNIT NUMBER 8.5 VECTORS 5 (Vector equations of straight lines) by A.J.Hobson 8.5.1 Introduction 8.5. The straight line passing through a given point and parallel to a given vector 8.5.3

### Pre-Calculus Review Problems Solutions

MATH 1110 (Lecture 00) August 0, 01 1 Algebra and Geometry Pre-Calculus Review Problems Solutions Problem 1. Give equations for the following lines in both point-slope and slope-intercept form. (a) The

### What are the place values to the left of the decimal point and their associated powers of ten?

The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything

### Algebra and Geometry Review (61 topics, no due date)

Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties

### North Carolina Math 2

Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4.

### Complex Numbers Basic Concepts of Complex Numbers Complex Solutions of Equations Operations on Complex Numbers

Complex Numbers Basic Concepts of Complex Numbers Complex Solutions of Equations Operations on Complex Numbers Identify the number as real, complex, or pure imaginary. 2i The complex numbers are an extension

### BX in ( u, v) basis in two ways. On the one hand, AN = u+

1. Let f(x) = 1 x +1. Find f (6) () (the value of the sixth derivative of the function f(x) at zero). Answer: 7. We expand the given function into a Taylor series at the point x = : f(x) = 1 x + x 4 x

### THREE DIMENSIONAL GEOMETRY

Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,

### GRE Prep: Precalculus

GRE Prep: Precalculus Franklin H.J. Kenter 1 Introduction These are the notes for the Precalculus section for the GRE Prep session held at UCSD in August 2011. These notes are in no way intended to teach

### LINEAR ALGEBRA W W L CHEN

LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,

### Math 497C Sep 9, Curves and Surfaces Fall 2004, PSU

Math 497C Sep 9, 2004 1 Curves and Surfaces Fall 2004, PSU Lecture Notes 2 15 sometries of the Euclidean Space Let M 1 and M 2 be a pair of metric space and d 1 and d 2 be their respective metrics We say

### by the matrix A results in a vector which is a reflection of the given

Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that

### www.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates

Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c

### Lecture 14: Section 3.3

Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in

### Parametric Equations and the Parabola (Extension 1)

Parametric Equations and the Parabola (Extension 1) Parametric Equations Parametric equations are a set of equations in terms of a parameter that represent a relation. Each value of the parameter, when

### Example SECTION 13-1. X-AXIS - the horizontal number line. Y-AXIS - the vertical number line ORIGIN - the point where the x-axis and y-axis cross

CHAPTER 13 SECTION 13-1 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants X-AXIS - the horizontal

### Inner Product Spaces

Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and

### 5.3 The Cross Product in R 3

53 The Cross Product in R 3 Definition 531 Let u = [u 1, u 2, u 3 ] and v = [v 1, v 2, v 3 ] Then the vector given by [u 2 v 3 u 3 v 2, u 3 v 1 u 1 v 3, u 1 v 2 u 2 v 1 ] is called the cross product (or

### Differentiation of vectors

Chapter 4 Differentiation of vectors 4.1 Vector-valued functions In the previous chapters we have considered real functions of several (usually two) variables f : D R, where D is a subset of R n, where

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### Core Maths C1. Revision Notes

Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the

### Prentice Hall Mathematics: Algebra 2 2007 Correlated to: Utah Core Curriculum for Math, Intermediate Algebra (Secondary)

Core Standards of the Course Standard 1 Students will acquire number sense and perform operations with real and complex numbers. Objective 1.1 Compute fluently and make reasonable estimates. 1. Simplify

### Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis

Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis 2. Polar coordinates A point P in a polar coordinate system is represented by an ordered pair of numbers (r, θ). If r >

### Geometry in a Nutshell

Geometry in a Nutshell Henry Liu, 26 November 2007 This short handout is a list of some of the very basic ideas and results in pure geometry. Draw your own diagrams with a pencil, ruler and compass where

### TOMS RIVER REGIONAL SCHOOLS MATHEMATICS CURRICULUM

Content Area: Mathematics Course Title: Precalculus Grade Level: High School Right Triangle Trig and Laws 3-4 weeks Trigonometry 3 weeks Graphs of Trig Functions 3-4 weeks Analytic Trigonometry 5-6 weeks

### NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS

NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

### DRAFT. Further mathematics. GCE AS and A level subject content

Further mathematics GCE AS and A level subject content July 2014 s Introduction Purpose Aims and objectives Subject content Structure Background knowledge Overarching themes Use of technology Detailed

### Elementary triangle geometry

Elementary triangle geometry Dennis Westra March 26, 2010 bstract In this short note we discuss some fundamental properties of triangles up to the construction of the Euler line. ontents ngle bisectors

### December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

### Lecture 7 : Inequalities 2.5

3 Lecture 7 : Inequalities.5 Sometimes a problem may require us to find all numbers which satisfy an inequality. An inequality is written like an equation, except the equals sign is replaced by one of

### POWER SETS AND RELATIONS

POWER SETS AND RELATIONS L. MARIZZA A. BAILEY 1. The Power Set Now that we have defined sets as best we can, we can consider a sets of sets. If we were to assume nothing, except the existence of the empty

### 1 Review of complex numbers

1 Review of complex numbers 1.1 Complex numbers: algebra The set C of complex numbers is formed by adding a square root i of 1 to the set of real numbers: i = 1. Every complex number can be written uniquely

### Dear Accelerated Pre-Calculus Student:

Dear Accelerated Pre-Calculus Student: I am very excited that you have decided to take this course in the upcoming school year! This is a fastpaced, college-preparatory mathematics course that will also

### Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year.

This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra

### Actually, if you have a graphing calculator this technique can be used to find solutions to any equation, not just quadratics. All you need to do is

QUADRATIC EQUATIONS Definition ax 2 + bx + c = 0 a, b, c are constants (generally integers) Roots Synonyms: Solutions or Zeros Can have 0, 1, or 2 real roots Consider the graph of quadratic equations.

### Solutions to Practice Problems

Higher Geometry Final Exam Tues Dec 11, 5-7:30 pm Practice Problems (1) Know the following definitions, statements of theorems, properties from the notes: congruent, triangle, quadrilateral, isosceles

### Linear Algebra Notes for Marsden and Tromba Vector Calculus

Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of

### ELEMENTS OF VECTOR ALGEBRA

ELEMENTS OF VECTOR ALGEBRA A.1. VECTORS AND SCALAR QUANTITIES We have now proposed sets of basic dimensions and secondary dimensions to describe certain aspects of nature, but more than just dimensions

### 4.3 Lagrange Approximation

206 CHAP. 4 INTERPOLATION AND POLYNOMIAL APPROXIMATION Lagrange Polynomial Approximation 4.3 Lagrange Approximation Interpolation means to estimate a missing function value by taking a weighted average

### Section 1.4. Lines, Planes, and Hyperplanes. The Calculus of Functions of Several Variables

The Calculus of Functions of Several Variables Section 1.4 Lines, Planes, Hyperplanes In this section we will add to our basic geometric understing of R n by studying lines planes. If we do this carefully,

### Estimated Pre Calculus Pacing Timeline

Estimated Pre Calculus Pacing Timeline 2010-2011 School Year The timeframes listed on this calendar are estimates based on a fifty-minute class period. You may need to adjust some of them from time to

### Equations and Inequalities

Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.

### Two vectors are equal if they have the same length and direction. They do not

Vectors define vectors Some physical quantities, such as temperature, length, and mass, can be specified by a single number called a scalar. Other physical quantities, such as force and velocity, must

### PYTHAGOREAN TRIPLES PETE L. CLARK

PYTHAGOREAN TRIPLES PETE L. CLARK 1. Parameterization of Pythagorean Triples 1.1. Introduction to Pythagorean triples. By a Pythagorean triple we mean an ordered triple (x, y, z) Z 3 such that x + y =

### MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education)

MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,

### ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form

ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola

### Introduction. The Aims & Objectives of the Mathematical Portion of the IBA Entry Test

Introduction The career world is competitive. The competition and the opportunities in the career world become a serious problem for students if they do not do well in Mathematics, because then they are

### 2x + y = 3. Since the second equation is precisely the same as the first equation, it is enough to find x and y satisfying the system

1. Systems of linear equations We are interested in the solutions to systems of linear equations. A linear equation is of the form 3x 5y + 2z + w = 3. The key thing is that we don t multiply the variables

### Vectors, Gradient, Divergence and Curl.

Vectors, Gradient, Divergence and Curl. 1 Introduction A vector is determined by its length and direction. They are usually denoted with letters with arrows on the top a or in bold letter a. We will use

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

### Week 13 Trigonometric Form of Complex Numbers

Week Trigonometric Form of Complex Numbers Overview In this week of the course, which is the last week if you are not going to take calculus, we will look at how Trigonometry can sometimes help in working