M427L Handout: Lines, Planes, Cross Products, & Coordinate Systems

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1 M427L Handout: Lines, Planes, Cross Products, & Coordinate Systems Salman Butt June 14, 2007 Exercises (1) Compute (2, 9, 3) ( 1, 2, 6). (2) Compute v 1, v 2 and v 1 v 2 for v 1 = ( 1, 3, 1) and v 2 = ( 2, 3, 7). (3) Find two nonparallel vectors both orthogonal to (1, 1, 1). Equations of Lines in R 3 We want to determine the equation of a line l parallel to a vector v 1 that passes through the endpoint of a vector v 2 : Often the best way to think of a line is to think of one parameter which varies, and as this parameter varies, we move along the line. We denote this parameter by t. Now traveling along the line l would be the same as traveling in the direction of v 1 since they are parallel. Thus consider 1

2 point of the form t v 1 for all real value of t which gives us a line that contains our vector v 1. But we want our line l to pass through the endpoint of v 2. If we look at the line t v 1, we see that we merely need to translate it over to the endpoint of v 2. How do we translate vectors:. Thus if we consider where t R, we see we have a line parallel to v 1 which passes through the endpoint of v 2 (just take t = 0). The equation of this line is given parametrically and indeed l(t) is a one-parameter family of vectors. In coordinates, this family of vectors is given by Equations of Planes in R 3 In order to specify a plane in R 3, we need two pieces of data: a point P 0 = (x 0, y 0, z 0 ) in the plane and a vector n = (A, B, C) normal to the plane. (Recall that we need two pieces of information to specify a line in R 2, a point on the line and its slope.) Now any arbitrary point P = (x, y, z) lies on the plane P determined by P 0 and n if and only if Why is that true? Consider the picture: Translating this into algebra, we find the point P is in the plane P if and only if And this equation precisely defines the plane P given by the point P 0 and n: the points in the plane are precisely those that satisfy the equation above. We can also restate this in the following handy way: This formulation allows us to easily read off the pertinent data when we see the equation of a plane given in its typical form: αx + βy + γz + δ = 0. 2

3 Linear Algebra and the Cross Product We will now discuss a second form of multiplication of vectors, the cross product. In order to give the formulation of the cross product, we will need just a little linear algebra. Recall the definition of 2 2 and 3 3 matrices and their determinants: It is important to note that our notion of the cross product does not generalize to higher dimensional space (there is a subtlety in what one means by plane ). We record a few basic properties of determinants of matrices: (1) (2) (3) 3

4 With these linear algebra facts in hand, we can now give the definition of the cross product. Given two vectors v 1 = (x 1, y 1, z 1 ), v 2 = (x 2, y 2, z 2 ), their cross (or vector) product v 1 v 2 is merely Expanding this, we have Let s quickly record some basic properties of the cross product: (1) (2) (3) What does the cross product mean geometrically? In order to describe this, we need to introduce some new ideas. Given three vectors v 1, v 2, v 3, consider their triple product defined as If we expand this out, we find But this is just the determinant of the matrix 4

5 Now some basic linear algebra tells us that if any row (or column) in a matrix is a linear combination of some other rows (or columns), then the determinant of that matrix is zero. Thus if the vector v 3 is in the plane spanned by v 1 and v 2 (this is precisely what it means for v 3 to be a linear combination of v 1 and v 2 ), then ( v 1 v 2 ) v 3 = 0. Recall that if for any two vectors a, b we have a b = 0, then the two vectors are orthogonal. Hence the vector v 1 v 2 is orthogonal to any vector in the plane spanned by v 1 and v 2, in particular to both v 1 and v 2. Now there are two vectors that are perpendicular to the plane spanned by v 1 and v 2 : Which is the correct vector? Well, by tradition, we use the right hand rule to orient our space and this rule also tells us which perpendicular vector to use: So now we know the direction of our vector v 1 v 2. But what about its length? Let s compute that: If we expand this last line and recollect terms, we find Recalling the dot product, we readily see that 5

6 A little trigonometry then tell us where 0 θ π is the angle between v 1 and v 2. If we take square roots, we find Note that the length of the vector is precisely the area of the parallelogram spanned by v 1 and v 2 : Thus we know precisely what the vector v 1 v 2 means geometrically. Coordinate Systems We have been considering exclusively Cartesian coordinates for R 3 so far: a point (or vector) is given by some triple (x, y, z) where x is the x-coordinate, y is the y-coordinate, and z is the z-coordinate. In one way, this system uses a box approach: it views three dimensional space as concentric boxes so that as you change each component of of (x, y, z), you travel along either the depth, width, or height of some box (or rectangle): But this is not the only perspective one can take. Consider another simple object, the cylinder. We can think of three-dimensional space as concentric cylinders of infinite height. If we want to specify a particular point, we pick one of these cylinders (specified by some radius r), where on this cylinder we are (specified by an angle θ, 0 θ < 2π), and at what height we are on the cylinder (specified by some real number z): 6

7 These coordinates (r, θ, z) are called cylindrical coordinates. One other way to think of these coordinates is as a generalization of polar coordinates in R 2 to R 3 : If we want to go from cylindrical coordinates to Cartesian coordinates, we can do so easily. Recalling how polar coordinates work, we readily see To go from Cartesian coordinates to cylindrical coordinates is also easy, though the formulas start to get more complicated. This essentialy has to do with determining our angle θ, but let us record the formula here: Cylindrical coordinates are handy to solve certain problems. The key feature of such problems is that there is some cylindrical symmetry (symmetry about a line). We will see such problems in the future so it is important to keep these coordinates in mind. One could solve such problems in Cartesian coordinates, but using cylindrical coordinates often simplify the problem greatly and make a seemingly intractable problem easy. Another model for three-dimensional space is a collection of concentric spheres. Here, we need to select a sphere (again specified by a radius, denoted ρ this time) and where on this sphere we are at (specified by two angles θ and φ with 0 θ < 2π and 0 φ π): 7

8 These coordinates (ρ, θ, φ) are called spherical coordinates and are another generalization of polar coordinates in R 2 to R 3. If we want to go from spherical coordinates to Cartesian coordinates, we can just study our picture above. Using a little geometry and trigonometry, we see Going from Cartesian coordinates to spherical coordinates is again the result of basic trigonometry: As a note, just be careful about taking these inverse trigonometry functions. Bear in mind that 0 θ < 2π and 0 φ π, so if you find that φ = 3π, φ got wound around too much and should instead be taken to be π: Finally, we should note that there exist unit vectors for each of these coordinate systems (just as we have the vectors i, j, k for Cartesian coordinates). See p. 72 for details, though we probably won t be using these very much. 8