C relative to O being abc,, respectively, then b a c.
|
|
- Rosalind Waters
- 8 years ago
- Views:
Transcription
1 2 EP-Program - Strisuksa School - Roi-et Math : Vectors Dr.Wattana Toutip - Department of Mathematics Khon Kaen University 200 :Wattana Toutip wattou@kku.ac.th 2. Vectors A quantity with both direction and magnitude is a vector. A quantity with magnitude only is a scalar. Fig 2. A vector with magnitude is a unit vector. x A two dimensional vector can be written as or as xy y, or as xi yi unit vectors in the x and y directions respectively. The magnitude or modulus of xy, is x, y x 2 y 2., where i and j are x Three dimensional vectors can be written as y or as x, y, z or as xi yj zk, where z i, j, k are unit vectors in the x, y, z directions respectively. The magnitude of x, y, x is x, y, z x 2 y 2 z Vector geometry If A and B are points, the vector from A to B is written AB. The vector from the origin to a point is the position vector of that point.
2 If the position vectors of A and B are a and b, then AB b a. Vector can be added by a vector triangle or a vector parallelogram. Note that if OABC is a parallelogram, with the position vectors of A, B, C relative to O being abc,, respectively, then b a c. Fig 2.2 One vector is parallel to another if is a scalar multiple of it i.e.: a is parallel to b if there is a scalar so that a b. If wa xb ya zb, where a and b are not parallel to each other, then w y and x z. If A has position vector a, then a general point on the line through A parallel to b has position vector a tb, where t is a scalar. 2.. Examples. OABC is a parallelogram, X and Y lie on OC and AC respectively, with 2 OX OC, AY AC. Show that X, Y, B lie on a straight line. 4 Fig 2. Solution Let the position vector relative to O of A and C be a and c. The position vector of B is a c. The position vector of X is 2 c. The position vector of Y is a c a a c YB a c a c a c
3 2 XB a c c a c. YB XB. It following that YB is parallel to XB. 4 X, Y, B lie on a straight line 2. OABC is a parallelogram, with the position vectors of A and C relative to O being a and c. X is the midpoint of OA, and OB and XC meet at Y. Find the position vector of Y. Solution Fig 2.4 As above, the position vector of B is a c.the position vector of X is 2 a. A general point on OB is t a c. A general point on XC is c s a c 2. Y is on both these lines. ta tc c sa sc. 2 This gives two equations: t s and t s. 2 2 Solve to find that s, t. Y has position vector a c 2..2 Exercises. Which of the following quantities are vectors and which are scalars? (a) Velocity (b) Mass (c) Magnetic field (d) Temperature 2. Find the magnitudes of the following vector.: (a),4 (b) 2,, 7
4 (c) i j (d) 2i j 2.5k. Find the vectors AB for the following points A and B : (a) A,7, B 4,9 (b) A2,7,4, B,2, (c) A x, y, B p, q (d) A2 x, y, z, Bx, y,4z 4. OAB is a triangle, and X and Y are the midpoint of OA and OB respectively. Let OA aand OB b. Find XY in terms of a and b. Show that XY is parallel to AB. 5. OAB is triangle with OA a and OB b. X and Y are such that OX a and OY b. If XY is parallel to AB show that. 6. OABC is a parallelogram, with OA a and OC c. Find OB in terms of a and c. What can you deduce? 7. A, B, C, D are four points in space,with position vectors a, b, c, d respectively. I, J, K, L are the midpoints of AB, BC, CD, DA respectively. Find the position vectors of I, J, K, L in terms of a, b, c, d. Find the vectors IJ, JK, LK, IL. What can you say about the figure IJKL? 8. A, B, C have position vectors abc,, respectively. Find the position vector of M, the midpoint of BC. G lies on AM with AG 2GM. Show that the position vector of G is a b c. Show that G lies also on the lines from B to the midpoint of AC and from C to the midpoint of AB. 9. The position vectors of A and B relative to O are a and b. X is the midpoint of OA and Y lies on OB with OY OB. XY meets AB at Z. Find the position vector of Z. 0. a and b are two non-parallel vectors. The position vectors of XY, and Z are a 2, b a b and a b.find the value of if XYand, Z lie on a straight 2 4 line.. ABCDEF is a regular hexagon, with centre at O.OA a and OB b. Express, in terms of a and b, FA, DE, CD, CF.
5 2.2 Scalar Product Fig 2.5 The scalar, or dot product of two vectors is defined as follows: a. b a b cos, Where is the angle between a and b. In terms of coordinates the scalar product is as follows: a, b. x, y ax by for two-dimensional vectors. ai bj ck xi yj zk ax by cz, for three-dimensional vectors. In particular, ab. 0if and only if a and b are perpendicular. In particular, putting a b, a. a a The component or projection of a in the direction of b is 2.2. Examples 2 ab. b. Find the angle between the vectors, 2, and 2, 4 Solution Apply the formula above., 2, 4, 2,. 2,, cos 8. 8 cos and. 2, A, B, C are the vertices of a triangle. Let the perpendiculars from A to BC and from B to AC meet at D. Show that CD is perpendicular to AB. (Note: D is called the orthocenter of ABC.) Solution Let A, B, C, D have position vectors a, b, c, d respectively. From A to BC and from information given : d a. b c 0 and d b a c. 0
6 Expand these: d. b d. c a. b a. c 0 d. a d. a b. c a. c 0 Subtract these equations: d. b d. a b. c a. c 0 d cb a 0 DC is perpendicular to AB. Fig Exercises. Evaluate the following: (a),2.,4 (b) 2, 4. 7, (c),2,5. 2,, (d) i 2 j. i 5 j 2. Find the angles between the pairs of vectors in Question.. Find the value of x if x, is perpendicular to 4. Find a vector which is perpendicular to 2,7. 5. Find a unit vector which is perpendicular to 7, 24,. 6. Find the values of x and y if xy,, is perpendicular to both, 2, 4 and 4, 2,. 7. Find a vector which is perpendicular to both,0, and 2,, Find a unit vector which is perpendicular to both,,5 and,,. 9. Find the component of 2, 4 in the direction of,. 0. Find the component of 2i j k in the direction of i j.. Verify that the component of xy, in the direction of,0 is x. 2. Verify that component of xi yj zk in the j direction is y.. Let OAB be a triangle, with OA a and OB b. Let c a b be the third side of the triangle. By expanding cc. prove the cosine rule for OAB. 4. OABC is a parallelogram, with OA a and Oc c. By considering OB. CA show that OABC is a rhombus if and only if its diagonals are perpendicular. 5. Let O be the circumcentre of ABC. Let the position vectors of A, B, C relative to O be abc.,, (So that a b c ). Show that ab c is the orthocenter of the triangle.(see (2) of 2.2.).
7 6. Find the point X on the line 2, t 4, such that OX is perpendicular to the line. Hence find the perpendicular distance from the origin to the line. 7. Find the point X on the line of Question 6 such that AX, where A is at 2, 7,is perpendicular to the line. Hence find the perpendicular distance from A to the line. 2. Lines and planes in three dimensions Suppose a line goes through a point vector a and is parallel to the vector parallel to the vector v. Then a general point on the line is : r a v Where is a scalar. If a ai bj ck and v vi uj wk, then the Cartesian equations of the line are : x a y b z c v u w Suppose a plane goes through a point with position vector a and is parallel to vector u and v. Then its equation can be written as : r. n a. n If n li mj nk, then the Cartesian equation of the plane is : lx my nz d Where d a. n is a constant. The shortest,i.e. perpendicular, distance from the point abc,, to the plane lx my nz d is : la mb nc d l m n The angle between two lines is the angle between their direction vectors. This can be found from the scalar product, defined in 2.2. To find the angle between a line and a plane, find the angle between the direction of the line and the perpendicular to the plane, and subtract that angle from 90. The angle between two planes is the angle between their perpendiculars 2.. Examples. Two lines are given by: 2 2 and r ti j 4k i j k r i j k i j Find the value of t so that the lines intersect. Find the angle between the lines. Solution Equate the values of r. i 2 j k 2i j ti j 4l i j k. This gives the following three equations:
8 2 t ;2 ; 4. From the last two equations and 2.Substitute these values into the first equation to find t. t 4 The angle between the lines is the angle between the two direction vectors. Find this by the scalar product. 5 cos. The angle is 9 2. Find the equation of the plane which contains the two lines of example. Solution Let a perpendicular to the plane be li mj nk. This vector is at right angles to the direction vectors of the lines. 2lm 0 and l m n. These equations are satisfied by l, m 2, n.hence a perpendicular vector is i 2 j k. The equation of the plane is x 2y z i 2 j k. i 2 j k The equation is x 2y z Exercises. Find vector equations for the following lines: (a) Through,, 2, parallel to 2i j 7k (b) Through, 0,, parallel to 4i 2j (c) Through,, and 2,5, 2 (d) Through,2, and, 2, 2. Find the Cartesian equations of the lines in Question.. Find the values of t so that the following pairs of lines intersect: (a) r,,2 ti j and r, 4, 2i j k x y 4 z 5 x y z (b) and t Find the angles between the pair of lines in Question. 5. Find in the form r a u v the equations of the following planes: (a) Through, 2,, parallel to i j k and i k. (b) Through,2,, 2,, and,,. 6. Find in the form r. n k the equation of the following planes: (a) Through, 2,,perpendicular to 2i j k (b) Through 2,, 4, perpendicular to the z -axis. (c) Through 2,, 4,parallel to i j and to k. (c) Through 2,,2, 2,,0 and 0, 4,.
9 7. Find the angles between the following: (a) The line 0,,2 i j and the plane r i j k (b) The line x 4 y z 2.. and the plane x y z 0 (c) The line 0,, 2i j k and the plane 8. Find the angles between the following pairs of planes : (a) x y 2z and 2x y z (b) r. i 2 j k 0 and r i j k. 2 (c) r,,0 i k j and 2,,7,, i j vk. r i j k 9. Find the perpendicular distance between the following: (a) The point,, and the plane x 2y z 4 (b) The point 2,0, and the plane 2x y 4z 0. Find the point where the line r 5, 7, i j 2k meets the plane x 2y z 4. Find the angle between the line and the plane.. Find the point where the line r, 0, i 2 j meets the plane 2x y z. Find the angle between the line and the plane. 2.4 Examination questions. Referred to the point O as origin, the position vectors of the points A and B are i j and 8i 4j respectively. The point P lies on the line OB, between O and B, such that 4OP OB. The point Q lies on the line AB, between A and B, such that 5AQ 2AB. Find, in term of i and j, the vectors OP, AB, AQ, OQ and AP. The lines AP and OQ intersect at the point R. Given that OR : OQ and AR : AP, express the position vector of R in terms of (a),i and j (b),i and j. Hence find the values of and. Prove that the length of OR is The position vectors, relative to an origin O, of three points A, B, C are 2i 2 j,5i j and i 9 j respectively. (i) Given that OB moa noc, where m and n are scalar constant, find the value of m and of n. (ii) Evaluate AB. BC and state the deduction which can be made about ABC. (iii)evaluate AB. BC and hence find BAC.. The position vectors of the points AB, with respect to an origin O are a and b respectively, where ABand, O are not collinear. Point L, M, N have position
10 vectors pa, qb and 2b a respectively, where p and q are non- constants and LM is not parallel to AB. Show that N lies on the line AB. Prove that L, M, N are collinear if 2p q p. If p and q, find LM LN. 2 If, further, a 2i j 6k and b 4i j 4k,find the position vector of the foot of the perpendicular from N to OA. 4. A Cartesian frame of reference, having origin O, is defined by the by mutually perpendicular unit vectors i, j and k. (i) Find the vector equation of the line which passes through the two points, having position relative to O, 2i j 2k and i j in the form r a tb. (ii) A second line has vector equation r i 4 j k s 2i j k 2 2. Find the point of intersection of the two lines. 5. Relative to an origin O,points A and B have position vectors 2i 9 j 6k and 6i j 6k respectively, i, j and k being orthogonal unit vectors. C is the point such that OC 2QA and D is the midpoint of AB. Find (i) The position vectors of C and D (ii) A vector equation of the line CD (iii)the angle AOB correct to the nearest degree. 6. Calculate unit vectors, or otherwise,calculate the equation of one of the bisectors of the angles between the lines By considering the sum of these two vectors, or otherwise, calculate the equation of one of the bisectors of the angles between the lines x s y 4 and x t y 4 Deduce the equation of one of the bisectors of the angles between the lines x 4 s y 4 and x 5 4 t, giving your answer in a similar form. y 9 7. The lines L and M have the equations r 2 s and 4 5 respectively. The plane has the equation 2 r r 4 t 2 6 2
11 (i) Verify that the point A with coordinates, 4,4 lies on L and on M but not on. (ii) Find the position vectors of the point of intersection B of L and. (iii)show that M and have no common point. 2 (iv) Find the cosine of the angle between the vectors and 0. 5 (v) Hence find, to the nearest degree, the angle between L and. 8. With respect to the origin O the points A, BC have position vectors 5, 4 4, 5 2 a i j k a i j k a i j k respectively, where a is a nonzero constant. Find (a) A vector equation for the line BC, (b) a vector equation for the plane OAB, (c) the cosine of the acute angle between the lines OA and OB. Obtain, in the form r. n p, a vector equation for, the plane which passes through A and is perpendicular to BC. Find Cartesian equation for (d) the plane, (e) the line BC. 9. The line L passes through A0,, and is parallel to 0, and the line M passes through B 2,, and is parallel to. 2 Show that L and M do not intersect. M ' is the line through A parallel to M. The lines M ' and L lie the plane. (i) Find the direction of the normal to. (ii) Find the angle between AB and. (iii)write the equation to in standard Cartesian form. (iv) Use the formula for the distance of a point from a plane to show that the shortest distance from B to is AB. Explain how you could have deduced this result from answer your answer to (ii).
12 Common errors. Points and Vectors (a) The vector from the origin to from say,7 to position. (b) The vector from 2,5 to,4 is,4. But this vector also goes 4,. Vectors have magnitude and direction, but not,0 is,5 and, 5. (c) In general, if A and B have position vectors a and b, then AB is b a. It is not b, or a b, or a b. 2. Modulus (a) The modulus of 2 2,4 is 4 5, not 7. (b) Be careful of negative signs. Any number squared is positive,so the modulus of, 4,4. is the same as the modulus of. Scalar Product (a) ab. is scalar, not a vector: 2,., 4. It should be (b) Do not be confused if there are many possible answers to a problem. There are infinitely many vectors perpendicular to, 2,,for example. Just give the simplest one you can find. 4. Lines and Planes (a) Do not confuse the position vector of a point on a line and the direction vector of that line. (b) In there dimension, two Cartesian equations are needed to define a line. If you have only one equation then that will describe a plane. (c) Do not confuse vectors perpendicular to a plane with vector parallel to the plane.
13 Solution (to exercise) ac, 2. (a) 5 (b) 54 (c) 0 (d) (a), 2 (b), 5, (c) p x, q y (d) x,2y, z 4. 2 b a 6. a c, a c 2 a b, b c, d a, c a, d b, c a, d b Parallelogram 8. 2 b c 9. 2a b b, a b, b,2a 2b (a) (b) 8 (c)9 (d) 7 2. (a)0. (b) 55. (c) 47.9 (d) , , x.8, y 5. 7., 4, ,, , 0.96, , 0.6,8.. (a),,2 2,,7 (b),0, 4,2,0 (c),,, 2, (d),2 0, 4,2
14 x y z 2 x y z 2. (a) (b) x y z (c) (d) x y 2 z (a) t (b) t 2 4. (a) 40.5 (b)98 5. (a), 2,,, (, 0,) (b),2,,,2, 0,,0 6. (a) r. 2,, (b) r. 0,0, 4 (c) r.,,0 0 (d) 7. (a) 25 (b)4 (c)7 8. (a) 60 (b) 50 (c) (a) 0.86 (b).67 0.,,.79.,,.47 2 =========================================================== References: Solomon, R.C. (997), A Level: Mathematics (4 th Edition), Great Britain, Hillman Printers(Frome) Ltd. r.,, 5
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,
More information5 VECTOR GEOMETRY. 5.0 Introduction. Objectives. Activity 1
5 VECTOR GEOMETRY Chapter 5 Vector Geometry Objectives After studying this chapter you should be able to find and use the vector equation of a straight line; be able to find the equation of a plane in
More informationVECTOR ALGEBRA. 10.1.1 A quantity that has magnitude as well as direction is called a vector. is given by a and is represented by a.
VECTOR ALGEBRA Chapter 10 101 Overview 1011 A quantity that has magnitude as well as direction is called a vector 101 The unit vector in the direction of a a is given y a and is represented y a 101 Position
More informationJUST 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
More informationMathematics Notes for Class 12 chapter 10. Vector Algebra
1 P a g e Mathematics Notes for Class 12 chapter 10. Vector Algebra A vector has direction and magnitude both but scalar has only magnitude. Magnitude of a vector a is denoted by a or a. It is non-negative
More informationSection 8.8. 1. The given line has equations. x = 3 + t(13 3) = 3 + 10t, y = 2 + t(3 + 2) = 2 + 5t, z = 7 + t( 8 7) = 7 15t.
. The given line has equations Section 8.8 x + t( ) + 0t, y + t( + ) + t, z 7 + t( 8 7) 7 t. The line meets the plane y 0 in the point (x, 0, z), where 0 + t, or t /. The corresponding values for x and
More informationFURTHER VECTORS (MEI)
Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level - MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: -9-7 Mathematics
More information1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two non-zero vectors u and v,
1.3. DOT PRODUCT 19 1.3 Dot Product 1.3.1 Definitions and Properties The dot product is the first way to multiply two vectors. The definition we will give below may appear arbitrary. But it is not. It
More informationa.) Write the line 2x - 4y = 9 into slope intercept form b.) Find the slope of the line parallel to part a
Bellwork a.) Write the line 2x - 4y = 9 into slope intercept form b.) Find the slope of the line parallel to part a c.) Find the slope of the line perpendicular to part b or a May 8 7:30 AM 1 Day 1 I.
More informationEquations Involving Lines and Planes Standard equations for lines in space
Equations Involving Lines and Planes In this section we will collect various important formulas regarding equations of lines and planes in three dimensional space Reminder regarding notation: any quantity
More informationwww.sakshieducation.com
LENGTH OF THE PERPENDICULAR FROM A POINT TO A STRAIGHT LINE AND DISTANCE BETWEEN TWO PAPALLEL LINES THEOREM The perpendicular distance from a point P(x 1, y 1 ) to the line ax + by + c 0 is ax1+ by1+ c
More informationL 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:
More informationDEFINITIONS. Perpendicular Two lines are called perpendicular if they form a right angle.
DEFINITIONS Degree A degree is the 1 th part of a straight angle. 180 Right Angle A 90 angle is called a right angle. Perpendicular Two lines are called perpendicular if they form a right angle. Congruent
More informationSelected practice exam solutions (part 5, item 2) (MAT 360)
Selected practice exam solutions (part 5, item ) (MAT 360) Harder 8,91,9,94(smaller should be replaced by greater )95,103,109,140,160,(178,179,180,181 this is really one problem),188,193,194,195 8. On
More informationSection 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
More informationCHAPTER 1. LINES AND PLANES IN SPACE
CHAPTER 1. LINES AND PLANES IN SPACE 1. Angles and distances between skew lines 1.1. Given cube ABCDA 1 B 1 C 1 D 1 with side a. Find the angle and the distance between lines A 1 B and AC 1. 1.2. Given
More informationSection 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,
More informationGCE Mathematics (6360) Further Pure unit 4 (MFP4) Textbook
Version 36 klm GCE Mathematics (636) Further Pure unit 4 (MFP4) Textbook The Assessment and Qualifications Alliance (AQA) is a company limited by guarantee registered in England and Wales 364473 and a
More informationVector Algebra CHAPTER 13. Ü13.1. Basic Concepts
CHAPTER 13 ector Algebra Ü13.1. Basic Concepts A vector in the plane or in space is an arrow: it is determined by its length, denoted and its direction. Two arrows represent the same vector if they have
More informationCIRCLE COORDINATE GEOMETRY
CIRCLE COORDINATE GEOMETRY (EXAM QUESTIONS) Question 1 (**) A circle has equation x + y = 2x + 8 Determine the radius and the coordinates of the centre of the circle. r = 3, ( 1,0 ) Question 2 (**) A circle
More informationCHAPTER FIVE. 5. Equations of Lines in R 3
118 CHAPTER FIVE 5. Equations of Lines in R 3 In this chapter it is going to be very important to distinguish clearly between points and vectors. Frequently in the past the distinction has only been a
More informationSection 9.5: Equations of Lines and Planes
Lines in 3D Space Section 9.5: Equations of Lines and Planes Practice HW from Stewart Textbook (not to hand in) p. 673 # 3-5 odd, 2-37 odd, 4, 47 Consider the line L through the point P = ( x, y, ) that
More informationA vector is a directed line segment used to represent a vector quantity.
Chapters and 6 Introduction to Vectors A vector quantity has direction and magnitude. There are many examples of vector quantities in the natural world, such as force, velocity, and acceleration. A vector
More informationMAT 1341: REVIEW II SANGHOON BAEK
MAT 1341: REVIEW II SANGHOON BAEK 1. Projections and Cross Product 1.1. Projections. Definition 1.1. Given a vector u, the rectangular (or perpendicular or orthogonal) components are two vectors u 1 and
More informationTwo 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
More informationProblem set on Cross Product
1 Calculate the vector product of a and b given that a= 2i + j + k and b = i j k (Ans 3 j - 3 k ) 2 Calculate the vector product of i - j and i + j (Ans ) 3 Find the unit vectors that are perpendicular
More informationExercise Set 3. Similar triangles. Parallel lines
Exercise Set 3. Similar triangles Parallel lines Note: The exercises marked with are more difficult and go beyond the course/examination requirements. (1) Let ABC be a triangle with AB = AC. Let D be an
More informationAdding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors
1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number
More informationProjective Geometry - Part 2
Projective Geometry - Part 2 Alexander Remorov alexanderrem@gmail.com Review Four collinear points A, B, C, D form a harmonic bundle (A, C; B, D) when CA : DA CB DB = 1. A pencil P (A, B, C, D) is the
More informationFigure 1.1 Vector A and Vector F
CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have
More informationEquation of a Line. Chapter H2. The Gradient of a Line. m AB = Exercise H2 1
Chapter H2 Equation of a Line The Gradient of a Line The gradient of a line is simpl a measure of how steep the line is. It is defined as follows :- gradient = vertical horizontal horizontal A B vertical
More informationDecember 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
More informationdiscuss how to describe points, lines and planes in 3 space.
Chapter 2 3 Space: lines and planes In this chapter we discuss how to describe points, lines and planes in 3 space. introduce the language of vectors. discuss various matters concerning the relative position
More informationSolutions 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
More informationAnalytical Geometry (4)
Analytical Geometry (4) Learning Outcomes and Assessment Standards Learning Outcome 3: Space, shape and measurement Assessment Standard As 3(c) and AS 3(a) The gradient and inclination of a straight line
More informationGeometry of Vectors. 1 Cartesian Coordinates. Carlo Tomasi
Geometry of Vectors Carlo Tomasi This note explores the geometric meaning of norm, inner product, orthogonality, and projection for vectors. For vectors in three-dimensional space, we also examine the
More informationProblem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday.
Math 312, Fall 2012 Jerry L. Kazdan Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday. In addition to the problems below, you should also know how to solve
More informationMathematics 205 HWK 6 Solutions Section 13.3 p627. Note: Remember that boldface is being used here, rather than overhead arrows, to indicate vectors.
Mathematics 205 HWK 6 Solutions Section 13.3 p627 Note: Remember that boldface is being used here, rather than overhead arrows, to indicate vectors. Problem 5, 13.3, p627. Given a = 2j + k or a = (0,2,
More informationInversion. Chapter 7. 7.1 Constructing The Inverse of a Point: If P is inside the circle of inversion: (See Figure 7.1)
Chapter 7 Inversion Goal: In this chapter we define inversion, give constructions for inverses of points both inside and outside the circle of inversion, and show how inversion could be done using Geometer
More informationSolutions 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
More informationSQA Higher Mathematics Unit 3
SCHOLAR Study Guide SQA Higher Mathematics Unit 3 Jane Paterson Heriot-Watt University Dorothy Watson Balerno High School Heriot-Watt University Edinburgh EH14 4AS, United Kingdom. First published 2001
More informationIMO Geomety Problems. (IMO 1999/1) Determine all finite sets S of at least three points in the plane which satisfy the following condition:
IMO Geomety Problems (IMO 1999/1) Determine all finite sets S of at least three points in the plane which satisfy the following condition: for any two distinct points A and B in S, the perpendicular bisector
More informationVector Notation: AB represents the vector from point A to point B on a graph. The vector can be computed by B A.
1 Linear Transformations Prepared by: Robin Michelle King A transformation of an object is a change in position or dimension (or both) of the object. The resulting object after the transformation is called
More informationSection 9-1. Basic Terms: Tangents, Arcs and Chords Homework Pages 330-331: 1-18
Chapter 9 Circles Objectives A. Recognize and apply terms relating to circles. B. Properly use and interpret the symbols for the terms and concepts in this chapter. C. Appropriately apply the postulates,
More information5.1 Midsegment Theorem and Coordinate Proof
5.1 Midsegment Theorem and Coordinate Proof Obj.: Use properties of midsegments and write coordinate proofs. Key Vocabulary Midsegment of a triangle - A midsegment of a triangle is a segment that connects
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Student Name:
GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, August 18, 2010 8:30 to 11:30 a.m., only Student Name: School Name: Print your name and the name of
More informationCo-ordinate Geometry THE EQUATION OF STRAIGHT LINES
Co-ordinate Geometry THE EQUATION OF STRAIGHT LINES This section refers to the properties of straight lines and curves using rules found by the use of cartesian co-ordinates. The Gradient of a Line. As
More informationIdentifying second degree equations
Chapter 7 Identifing second degree equations 7.1 The eigenvalue method In this section we appl eigenvalue methods to determine the geometrical nature of the second degree equation a 2 + 2h + b 2 + 2g +
More informationSan Jose Math Circle April 25 - May 2, 2009 ANGLE BISECTORS
San Jose Math Circle April 25 - May 2, 2009 ANGLE BISECTORS Recall that the bisector of an angle is the ray that divides the angle into two congruent angles. The most important results about angle bisectors
More information11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space
11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of
More informationVector Algebra II: Scalar and Vector Products
Chapter 2 Vector Algebra II: Scalar and Vector Products We saw in the previous chapter how vector quantities may be added and subtracted. In this chapter we consider the products of vectors and define
More informationQUADRILATERALS CHAPTER 8. (A) Main Concepts and Results
CHAPTER 8 QUADRILATERALS (A) Main Concepts and Results Sides, Angles and diagonals of a quadrilateral; Different types of quadrilaterals: Trapezium, parallelogram, rectangle, rhombus and square. Sum of
More information... ... . (2,4,5).. ...
12 Three Dimensions ½¾º½ Ì ÓÓÖ Ò Ø ËÝ Ø Ñ So far wehave been investigatingfunctions ofthe form y = f(x), withone independent and one dependent variable Such functions can be represented in two dimensions,
More informationUnified Lecture # 4 Vectors
Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1982, 2008. This chapter originates from material used by author at Imperial College, University of London, between 1981 and 1990. It is available free to all individuals,
More informationCHAPTER 8 QUADRILATERALS. 8.1 Introduction
CHAPTER 8 QUADRILATERALS 8.1 Introduction You have studied many properties of a triangle in Chapters 6 and 7 and you know that on joining three non-collinear points in pairs, the figure so obtained is
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 13, 2015 8:30 to 11:30 a.m., only.
GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, August 13, 2015 8:30 to 11:30 a.m., only Student Name: School Name: The possession or use of any communications
More informationREVIEW OF ANALYTIC GEOMETRY
REVIEW OF ANALYTIC GEOMETRY The points in a plane can be identified with ordered pairs of real numbers. We start b drawing two perpendicular coordinate lines that intersect at the origin O on each line.
More information12.5 Equations of Lines and Planes
Instructor: Longfei Li Math 43 Lecture Notes.5 Equations of Lines and Planes What do we need to determine a line? D: a point on the line: P 0 (x 0, y 0 ) direction (slope): k 3D: a point on the line: P
More information28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE. v x. u y v z u z v y u y u z. v y v z
28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.4 Cross Product 1.4.1 Definitions The cross product is the second multiplication operation between vectors we will study. The goal behind the definition
More informationTHEORETICAL MECHANICS
PROF. DR. ING. VASILE SZOLGA THEORETICAL MECHANICS LECTURE NOTES AND SAMPLE PROBLEMS PART ONE STATICS OF THE PARTICLE, OF THE RIGID BODY AND OF THE SYSTEMS OF BODIES KINEMATICS OF THE PARTICLE 2010 0 Contents
More informationExample 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
More informationSQA Advanced Higher Mathematics Unit 3
SCHOLAR Study Guide SQA Advanced Higher Mathematics Unit 3 Jane S Paterson Heriot-Watt University Dorothy A Watson Balerno High School Heriot-Watt University Edinburgh EH14 4AS, United Kingdom. First published
More information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More information13.4 THE CROSS PRODUCT
710 Chapter Thirteen A FUNDAMENTAL TOOL: VECTORS 62. Use the following steps and the results of Problems 59 60 to show (without trigonometry) that the geometric and algebraic definitions of the dot product
More information9 Multiplication of Vectors: The Scalar or Dot Product
Arkansas Tech University MATH 934: Calculus III Dr. Marcel B Finan 9 Multiplication of Vectors: The Scalar or Dot Product Up to this point we have defined what vectors are and discussed basic notation
More informationMath 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)
More information9.4. The Scalar Product. Introduction. Prerequisites. Learning Style. Learning Outcomes
The Scalar Product 9.4 Introduction There are two kinds of multiplication involving vectors. The first is known as the scalar product or dot product. This is so-called because when the scalar product of
More informationConcepts in Calculus III
Concepts in Calculus III Beta Version UNIVERSITY PRESS OF FLORIDA Florida A&M University, Tallahassee Florida Atlantic University, Boca Raton Florida Gulf Coast University, Ft. Myers Florida International
More information5.3 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
More informationShape, Space and Measure
Name: Shape, Space and Measure Prep for Paper 2 Including Pythagoras Trigonometry: SOHCAHTOA Sine Rule Cosine Rule Area using 1-2 ab sin C Transforming Trig Graphs 3D Pythag-Trig Plans and Elevations Area
More informationIncenter Circumcenter
TRIANGLE: Centers: Incenter Incenter is the center of the inscribed circle (incircle) of the triangle, it is the point of intersection of the angle bisectors of the triangle. The radius of incircle is
More informationSection 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.
More information= y y 0. = z z 0. (a) Find a parametric vector equation for L. (b) Find parametric (scalar) equations for L.
Math 21a Lines and lanes Spring, 2009 Lines in Space How can we express the equation(s) of a line through a point (x 0 ; y 0 ; z 0 ) and parallel to the vector u ha; b; ci? Many ways: as parametric (scalar)
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Tuesday, August 13, 2013 8:30 to 11:30 a.m., only.
GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Tuesday, August 13, 2013 8:30 to 11:30 a.m., only Student Name: School Name: The possession or use of any communications
More informationLINES AND PLANES CHRIS JOHNSON
LINES AND PLANES CHRIS JOHNSON Abstract. In this lecture we derive the equations for lines and planes living in 3-space, as well as define the angle between two non-parallel planes, and determine the distance
More informationVisualizing Triangle Centers Using Geogebra
Visualizing Triangle Centers Using Geogebra Sanjay Gulati Shri Shankaracharya Vidyalaya, Hudco, Bhilai India http://mathematicsbhilai.blogspot.com/ sanjaybhil@gmail.com ABSTRACT. In this paper, we will
More informationBaltic Way 1995. Västerås (Sweden), November 12, 1995. Problems and solutions
Baltic Way 995 Västerås (Sweden), November, 995 Problems and solutions. Find all triples (x, y, z) of positive integers satisfying the system of equations { x = (y + z) x 6 = y 6 + z 6 + 3(y + z ). Solution.
More informationLecture 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
More informationAP Physics - Vector Algrebra Tutorial
AP Physics - Vector Algrebra Tutorial Thomas Jefferson High School for Science and Technology AP Physics Team Summer 2013 1 CONTENTS CONTENTS Contents 1 Scalars and Vectors 3 2 Rectangular and Polar Form
More informationhttp://www.castlelearning.com/review/teacher/assignmentprinting.aspx 5. 2 6. 2 1. 10 3. 70 2. 55 4. 180 7. 2 8. 4
of 9 1/28/2013 8:32 PM Teacher: Mr. Sime Name: 2 What is the slope of the graph of the equation y = 2x? 5. 2 If the ratio of the measures of corresponding sides of two similar triangles is 4:9, then the
More informationStraight Line. Paper 1 Section A. O xy
PSf Straight Line Paper 1 Section A Each correct answer in this section is worth two marks. 1. The line with equation = a + 4 is perpendicular to the line with equation 3 + + 1 = 0. What is the value of
More informationPOINT OF INTERSECTION OF TWO STRAIGHT LINES
POINT OF INTERSECTION OF TWO STRAIGHT LINES THEOREM The point of intersection of the two non parallel lines bc bc ca ca a x + b y + c = 0, a x + b y + c = 0 is,. ab ab ab ab Proof: The lines are not parallel
More informationExam 1 Sample Question SOLUTIONS. y = 2x
Exam Sample Question SOLUTIONS. Eliminate the parameter to find a Cartesian equation for the curve: x e t, y e t. SOLUTION: You might look at the coordinates and notice that If you don t see it, we can
More informationBALTIC OLYMPIAD IN INFORMATICS Stockholm, April 18-22, 2009 Page 1 of?? ENG rectangle. Rectangle
Page 1 of?? ENG rectangle Rectangle Spoiler Solution of SQUARE For start, let s solve a similar looking easier task: find the area of the largest square. All we have to do is pick two points A and B and
More informationSIMSON S THEOREM MARY RIEGEL
SIMSON S THEOREM MARY RIEGEL Abstract. This paper is a presentation and discussion of several proofs of Simson s Theorem. Simson s Theorem is a statement about a specific type of line as related to a given
More information1. Vectors and Matrices
E. 8.02 Exercises. Vectors and Matrices A. Vectors Definition. A direction is just a unit vector. The direction of A is defined by dir A = A, (A 0); A it is the unit vector lying along A and pointed like
More informationAlgebra III. Lesson 33. Quadrilaterals Properties of Parallelograms Types of Parallelograms Conditions for Parallelograms - Trapezoids
Algebra III Lesson 33 Quadrilaterals Properties of Parallelograms Types of Parallelograms Conditions for Parallelograms - Trapezoids Quadrilaterals What is a quadrilateral? Quad means? 4 Lateral means?
More informationOne advantage of this algebraic approach is that we can write down
. Vectors and the dot product A vector v in R 3 is an arrow. It has a direction and a length (aka the magnitude), but the position is not important. Given a coordinate axis, where the x-axis points out
More informationLecture 24: Saccheri Quadrilaterals
Lecture 24: Saccheri Quadrilaterals 24.1 Saccheri Quadrilaterals Definition In a protractor geometry, we call a quadrilateral ABCD a Saccheri quadrilateral, denoted S ABCD, if A and D are right angles
More informationAngles in a Circle and Cyclic Quadrilateral
130 Mathematics 19 Angles in a Circle and Cyclic Quadrilateral 19.1 INTRODUCTION You must have measured the angles between two straight lines, let us now study the angles made by arcs and chords in a circle
More informationGeometry: Unit 1 Vocabulary TERM DEFINITION GEOMETRIC FIGURE. Cannot be defined by using other figures.
Geometry: Unit 1 Vocabulary 1.1 Undefined terms Cannot be defined by using other figures. Point A specific location. It has no dimension and is represented by a dot. Line Plane A connected straight path.
More informationVector Algebra and Calculus
Vector Algebra and Calculus 1. Revision of vector algebra, scalar product, vector product 2. Triple products, multiple products, applications to geometry 3. Differentiation of vector functions, applications
More informationUnited Arab Emirates University College of Sciences Department of Mathematical Sciences HOMEWORK 1 SOLUTION. Section 10.1 Vectors in the Plane
United Arab Emirates University College of Sciences Deartment of Mathematical Sciences HOMEWORK 1 SOLUTION Section 10.1 Vectors in the Plane Calculus II for Engineering MATH 110 SECTION 0 CRN 510 :00 :00
More information4. How many integers between 2004 and 4002 are perfect squares?
5 is 0% of what number? What is the value of + 3 4 + 99 00? (alternating signs) 3 A frog is at the bottom of a well 0 feet deep It climbs up 3 feet every day, but slides back feet each night If it started
More information1 Introduction to Matrices
1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns
More informationMATH 275: Calculus III. Lecture Notes by Angel V. Kumchev
MATH 275: Calculus III Lecture Notes by Angel V. Kumchev Contents Preface.............................................. iii Lecture 1. Three-Dimensional Coordinate Systems..................... 1 Lecture
More information2.1. Inductive Reasoning EXAMPLE A
CONDENSED LESSON 2.1 Inductive Reasoning In this lesson you will Learn how inductive reasoning is used in science and mathematics Use inductive reasoning to make conjectures about sequences of numbers
More informationLINES AND PLANES IN R 3
LINES AND PLANES IN R 3 In this handout we will summarize the properties of the dot product and cross product and use them to present arious descriptions of lines and planes in three dimensional space.
More information9 MATRICES AND TRANSFORMATIONS
9 MATRICES AND TRANSFORMATIONS Chapter 9 Matrices and Transformations Objectives After studying this chapter you should be able to handle matrix (and vector) algebra with confidence, and understand the
More information