St Andrew s Academy Mathematics Department Higher Mathematics VECTORS

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1 St Andrew s Academy Mathematics Department Higher Mathematics VECTORS

2 hsn.uk.net Higher Mathematics Vectors Contents Vectors 1 1 Vectors and Scalars EF 1 Components EF 1 Magnitude EF 4 Equal Vectors EF 4 5 Addition and Subtraction of Vectors EF 5 6 Multiplication by a Scalar EF 7 7 Position Vectors EF 8 8 Basis Vectors EF 9 9 Collinearity EF Dividing Lines in a Ratio EF The Scalar Product EF 14 1 The Angle Between Vectors EF 17 1 Perpendicular Vectors EF 0 14 Properties of the Scalar Product EF 1 CfE Edition This document was produced specially for the HSN.uk.net website, and we require that any copies or derivative works attribute the work to Higher Still Notes. For more details about the copyright on these notes, please see

3 Vectors Vectors 1 Vectors and Scalars EF A scalar is a quantity with magnitude (size) only for example, an amount of money or a length of time. Sometimes size alone is not enough to describe a quantity for example, directions to the nearest shop. For this we need to know a magnitude (i.e. how far), and a direction. Quantities with magnitude and direction are called vectors. A vector is named either by using the letters at the end of a directed line segment (e.g. AB represents a vector starting at point A and ending at point B) or by using a bold letter (e.g. u). You will see bold letters used in printed text, but in handwriting you should just underline the letter. B A Throughout these notes, we will show vectors in bold as well us underlining them (e.g. u). Components EF A vector may be represented by its components, which we write in a column. For example, is a vector in two dimensions. In this case, the first component is and this tells us to move units in the x-direction. The second component tells us to move units in the y-direction. So if the vector starts at the origin, it will look like: y u O x hsn.uk.net Page 1 CfE Edition

4 Vectors Note that we write the components in a column to avoid confusing them with coordinates. The following diagram also shows the vector, but in this case it does not start at the origin. y 1, 1, 1 O x Vectors in Three Dimensions In a vector with three components, the first two tell us ho many units to move in the x- and y-directions, as before. The third component specifies how far to move in the z-direction. z When looking at a pair of x, y -axes, the z-axis points out of the page from the origin. A set of D axes can be drawn on a page as shown to the right. z y O For example, 4 1 is a vector in three dimensions. This vector is shown in the diagram, starting from the origin. z O 4 y 1 x x Zero Vectors Any vector with all components zero is called a zero vector and can be written 0 as 0, e.g hsn.uk.net Page CfE Edition

5 Vectors Magnitude EF The magnitude (or length) of a vector u is written as u. It can be calculated as follows. EXAMPLES 1. Given u If If 5 u 1, find u units.. Find the length of a units. Unit Vectors a u b then u a b a u b then u a b c c 5 a 6. Any vector with a magnitude of one is called a unit vector. For example: if 1 0 u then u 1 0 So u is a unit vector unit. hsn.uk.net Page CfE Edition

6 Vectors Distance in Three Dimensions The distance between the points A and B is d AB AB units. For example, given 1 AB 5, we find d AB In fact, there is a three-dimensional version of the distance formula. The distance d between the points EXAMPLE x y z and,, 1, 1, x y z is d x x y y z z units. Find the distance between the points 1, 4,1 and 0,5, The distance is x x y y z z units. 4 Equal Vectors EF Vectors with the same magnitude and direction are equal. For example, all the vectors shown to the right are equal. If vectors are equal to each other, then all of their components are equal, i.e. p r q s t a d if b e then a d, b e and c f. c f Conversely, two vectors are only equal if all of their components are equal. hsn.uk.net Page 4 CfE Edition

7 Vectors 5 Addition and Subtraction of Vectors EF Consider the following vectors: a b c Addition We can construct a b as follows: b a a b a b means a followed by b. Similarly, we can construct a b c as follows: a a b c means a followed by b followed by c. To add vectors, we position them nose-to-tail. Then the sum of the vectors is the vector between the first tail and the last nose. Subtraction Now consider we can use vector addition to obtain b a b c c a b. This can be written as a b. a b, so if we first find b, b b is just b but in the opposite direction. b b and b have the same magnitude, i.e. b b. Therefore we can construct b a b as follows: a b a a b means a followed by b. hsn.uk.net Page 5 CfE Edition

8 Vectors Using Components If we have the components of vectors, then things become much simpler. The following rules can be used for addition and subtraction. a d a d b e b e c f c f add the components a d a d b e b e c f c f subtract the components EXAMPLES 1. Given 1 u 5 and 1 1 u v Given 4 p and 4 1 p q v, calculate u v and u v u v q 6, calculate p q and q p q p hsn.uk.net Page 6 CfE Edition

9 Vectors 6 Multiplication by a Scalar EF A vector u which is multiplied by a scalar k 0 will give the result ku. This vector will be k times as long, i.e. the magnitude will be k u. Note that if k 0 this means that the vector ku will be in the opposite direction to u. For example: u u u 1 u a ka If u b then ku kb. c kc Each component is multiplied by the scalar. EXAMPLES 1. Given 1 v 5, find v. 1 v Given 6 r, find 4r r hsn.uk.net Page 7 CfE Edition

10 Vectors Negative Vectors The negative of a vector is the vector multiplied by 1. If we write a vector as a directed line segment AB, then AB BA: AB A B A B AB BA 7 Position Vectors EF OA is called a position vector of point A relative to the origin O, and is written as a. OB is called the position vector of point B, written b. z y Given P x, yz,, the position vector OP or P x p has components y. z O x y O a A b B x To move from point A to point B we can move back along the vector a to the origin, and along vector b to point B. AB AO OB OA OB ab b a For the vector joining any two points P and Q, PQ q p. hsn.uk.net Page 8 CfE Edition

11 Vectors EXAMPLE R is the point,, From the coordinates, RS s r and S is the point 4, 6, 1 r and 4 s Find RS. 8 Basis Vectors EF A vector may also be defined in terms of the basis vectors i, j and k. These are three mutually perpendicular unit vectors (i.e. they are perpendicular to each other). These basis vectors can be written in component form as 1 i 0, 0 0 j 1 and 0 0 k 0. 1 Any vector can be written in basis form using i, j and k. For example: i j 6k There is no need for the working above if the following is used: a ai bj ck b. c k i j hsn.uk.net Page 9 CfE Edition

12 Vectors 9 Collinearity EF In Straight Lines, we learned that points are collinear if they lie on the same straight line. The points A, B and C in D space are collinear if AB is parallel to BC, with B a common point. Note that we cannot find gradients in three dimensions instead we use the following. Non-zero vectors are parallel if they are scalar multiples of the same vector. For example: If If EXAMPLE u 1 and p 9 and 6 6 v 1u then u and v are parallel q 1 4 then p and q are parallel. 8 A is the point 1,, 5, B8, 5,9 and C, 11,17. Show that A, B and C are collinear. AB b a BC c b BC AB, so AB and BC are parallel and since B is a common point, A, B and C are collinear. hsn.uk.net Page 10 CfE Edition

13 Vectors 10 Dividing Lines in a Ratio EF There is a simple process for finding the coordinates of a point which divides a line segment in a given ratio. EXAMPLE 1. P is the point, 4, 1 and R is the point 8, 1,19. The point T divides PR in the ratio :. Find the coordinates of T. Step 1 Make a sketch of the line, showing the ratio in which the point divides the line segment. Step Using the sketch, equate the ratio of the two lines with the given ratio. Step Cross multiply, then change directed line segments to position vectors. Step 4 Rearrange to give the position vector of the unknown point. Step 5 From the position vector, state the coordinates of the unknown point. R PT TR PT TR t p r t T t p r t t t r p 8 5t t t 10 5 t 7 So T is the point,, 7. P hsn.uk.net Page 11 CfE Edition

14 Vectors Using the Section Formula The previous method can be condensed into a formula as shown below. If the point P divides the line AB in the ratio m: n, then: na mb p, n m where is a, b and p are the position vectors of A, B and P respectively. It is not necessary to know this, since the approach explained above will always work. EXAMPLE. P is the point, 4, 1 and R is the point 8, 1,19. The point T divides PR in the ratio :. Find the coordinates of T. The ratio is :, so let m and n, then: np mr t n m p r So T is the point,, 7. Note If you are confident with arithmetic, this step can be done mentally. hsn.uk.net Page 1 CfE Edition

15 Vectors Further Examples EXAMPLES. The cuboid OABCDEFG is shown in the diagram. E F H D A G B O C The point A has coordinates 0,0,5, C8,0,0 and G8,1,0. The point H divides BF in the ratio 4:1. Find the coordinates of H. From the diagram: OH OA AB 4 5 BF OA OC 4 5 CG h a c 4 5 g c a c g 5c a 1 4 5c 5 g So H has coordinates 8, 48 5,5. 4. The points P6,1,, Q8,,1 and R9, 5, are collinear. Find the ratio in which Q divides PR. Since the points are collinear PQ kqr for some k. Working with the first components: 86k 98 k. Therefore PQ QR so Q divides PR in the ratio :1. Note BH 4 BF 5, so BH 4 5 BF. Note The ratio is :1 since PQ QR 1. hsn.uk.net Page 1 CfE Edition

16 Vectors 5. The points A 7, 4, 4, B1,5, 7 and C are collinear. Given that B divides AC in the ratio :, find the coordinates of C. AB 5 AC b a 5 c a to see this: b a 5c 5a 5c b 5a A 5 c b a So C has coordinates 17,11, 9. Note A sketch may help you B C 11 The Scalar Product EF So far we have added and subtracted vectors and multiplied a vector by a scalar. Now we will consider the scalar product, which is a form of vector multiplication. The scalar product is denoted by ab. (sometimes it is called the dot product) and can be calculated using the formula: ab. a b cosθ, where θ is the angle between the two vectors a and b. This formula is given in the exam. hsn.uk.net Page 14 CfE Edition

17 Vectors The definition above assumes that the vectors a and b are positioned so that they both point away from the angle, or both point into the angle. a θ b a θ b However, if one vector is pointing away from the angle, while the other points into the angle, a θ b a θ b we find that ab. a b cosθ. EXAMPLES 1. Two vectors, a and b have magnitudes 7 and units respectively and are at an angle of 60 to each other as shown below. b 60 What is the value of ab?. a ab. a bcosθ 7cos The vector u has magnitude k and v is twice as long as u. The angle between u and v is 0, as shown below. v Find an expression for uv. in terms of k. uv. u vcosθ kkcos0 k. k 0 u Remember When one vector points in and one points out, uv. u v cosθ. hsn.uk.net Page 15 CfE Edition

18 Vectors The Component Form of the Scalar Product The scalar product can also be calculated as follows: ab. ab ab ab where This is given in the exam. 1 1 a1 a a and a b1 b b b EXAMPLES. Find pq,. given that pq. pq pq pq p and If A is the point AB.AC. q.,, 9, B1,4, and C 1,, 6, calculate C 1,, 6 We need to use the position vectors of the points: B1,4, AB b a AC c a A,, AB.AC hsn.uk.net Page 16 CfE Edition

19 Vectors 1 The Angle Between Vectors EF The formulae for the scalar product can be rearranged to give the following equations, both of which can be used to calculate θ, the angle between two vectors.. cosθ ab ab ab ab ab cosθ ab or 1 1. Look back to the formulae for finding the scalar product, given on the previous pages. Notice that the first equation is simply a rearranged form of the one which can be used to find the scalar product. Also notice that the second simply replaces ab. with the component form of the scalar product. These formulae are not given in the exam but can both be easily derived from the formulae on the previous pages (which are given in the exam). EXAMPLES 1. Calculate the angle θ between vectors p i 4j k and q 4i j k. 4 p 4 and q 1 pq 1 1 pq pq cosθ pq θ cos (to 1 d.p.) (or radians (to d.p.)) hsn.uk.net Page 17 CfE Edition

20 Vectors. K is the point 1, 7, Start with a sketch: L,,4 θ, L,,4 and M,5,1 M,5,1. Find KLM. K1, 7, Now find the vectors pointing away from the angle: 1 4 LK k l 7 10, 4 5 LM ml Use the scalar product to find the angle: LK.LM cosθ LK LM θ cos (to 1 d.p.) (or radians (to d.p.)) hsn.uk.net Page 18 CfE Edition

21 Vectors. The diagram below shows the cube OPQRSTUV. The point R has coordinates 4,0,0. (a) Write down the coordinates of T and U. (b) Find the components of RT and RU. (c) Calculate the size of angle TRU. z S O (a) From the diagram, T0,4,4 and U4,4,4. P T V R U x Q y (b) RT t r 40 4, RU u r RT.RU (c) cos TRU RT RU cos TRU 6 5. (to 1 d.p.) (or radians (to d.p.)) hsn.uk.net Page 19 CfE Edition

22 Vectors 1 Perpendicular Vectors EF If a and b are perpendicular then ab. 0. This is because ab. a bcosθ abcos90 ( θ 90 since perpendicular) 0 (since cos900) Conversely, if ab. 0 then a and b are perpendicular. EXAMPLES 1. Two vectors are defined as a 4i j 5k and b i j k. Show that a and b are perpendicular. ab. ab 1 1ab ab Since ab. 0, a and b are perpendicular.. 4 PQ a 7 and RS a and RS where a is a constant. Given that PQ Since PQ and RS are perpendicular, PQ.RS 0 4 a 7a 0 8a7a 0 84a 0 a. are perpendicular, find the value of a. hsn.uk.net Page 0 CfE Edition

23 Vectors 14 Properties of the Scalar Product EF Some properties of the scalar product are as follows: ab. ba. a. b c ab. ac. (Expanding brackets) aa. a Note that these are not given in the exam. EXAMPLES 1. In the diagram, Calculate p. q r. p, r 4 and q. p q r p. q r p. q p. r pqcosθ1 prcosθ cos604cos In the diagram below a c and b. a c 0 a a b c. Calculate. a. a b c aa. ab. ac. a a b cosθ1 a c cosθ cos0 cos b 0 Remember ac. a c cosθ since a points to θ and c points away. hsn.uk.net Page 1 CfE Edition

36 Vectors 1. a = 1-4 and b = 0. Calculate (a) b (b) a - b (c) ( a+ b). (a) Find the magnitude of the vector 1. (b) Find a vector parallel to the vector which has unit length.. A is (0,-,5), B is (7,-6,9) and C is (1,-1,17). Show that A, B and C are collinear stating the ratio AB:BC. 4. PQRS is a parallelogram with P(,4,0), Q(7,6,-) and R(8,5,). Find the coordinates of S. 5. (a) P is the point (-1,8,0) and Q is (4,-,5). B divides PQ in the ratio :. Find the coordinates of B. (b) A is (0,1,5) and C is (8,5,-). Show that A, B and C are collinear. 6. An aeroplane flies in a straight line at a constant speed. It takes hours to fly from A to B and 4 hours to fly from B to C. Relative to coordinate axes, A is (0,-1,6) and C is (7,6,-1). Find the coordinates of B. 7. u = i j + 4k and v = i + aj + 7 k. If u = v find the value of a. 8. Show that the vectors a = i 4j + 6k and b = 4i 7j 6k are perpendicular. 9. A triangle has vertices A(6,-1,9), B(,-,11) and C(7,-8,14). Show that this triangle is right-angled at B.

37 10. Three points A, B and D have coordinates as shown. A(-1,0,4) C D(1,-8,) B(5,-6,16) (a) Find the coordinates of C if AB is parallel and equal in length to CD. (b) The point E divides AB in the ratio :1, find the coordinates of E. (c) Prove that CE is perpendicular to AB. 11. Use the diagrams to find the value of a.b. (a) a (b) a 60 0 b π 6 b a = 6 b = 7 a = 4 b = 1. Write down the value of p.q. p = 8 q = 9 q p 1. A triangle is formed from R(0,4,-1), S(1,5,) and T(6,1,-). (a) Find the vectors RS and RT. (b) Evaluate RS. RT (c) What can you deduce about he lines RS and RT. 14. A, B, C and D are the points (-1,,1), (1,6,7), (0,,5) and (1,4,10) respectively. (a) Find the components of AB and CD. p (b) The vector q is perpendicular to both AB 1 and CD. Find p and q.

38 u = and v = 5. k -1 (a) Write down the vectors u + v and u v. (b) Given that u + v and u v are perpendicular find k. 16. In the square based pyramid opposite all eight edges are of length 5 units. V Evaluate p.(q + r). C D B A 17. Shown opposite is a right-angled isosceles triangle. The two equal sides of the triangle have length 4 units. Find the value of k.(h + k + l). k h l T 18. In the diagram opposite TOPQR is a pyramid whose base OPQR is a rhombus of length 1 unit. OPT and ORT are equilateral triangles. R t (a) Evaluate t.r. (b) Given X is the midpoint of PQ, evaluate t.x. r O p 19. The diagram shows two vectors a and b with = and b =. (a) Evaluate (a) a.a (b) b.b (c) a.b a (b) Given p = a + b evaluate p.p. x P X Q 45 0 b

39 0. In the trapezium AB = DC and AB is paralle to DC. In terms of u and v, write down the vectors (a) AB (b) AC (c) BC (d) AN D u C _. N _ A B 1. ABCDEFGH is a parallelipiped. In terms of u, v and w find expressions for (a) DC (b) HC (c) AC (d) FD (e) CF u B C w A v D G H F E. (a) For the diagram opposite find AS and AT. (b) Hence calculate angle TAS. A(1,,-1) S(5,5,) T(4,0,) E(-1,0,-). Calculate the size of angle FEG in the diagram shown. F(1,1,0) G(0,,) 4. PA and PB are representatives of the vectors a and b. 4 - a = - 4 and b= and angle APB = θ. 1 a A 7 (a) Prove that cos θ = 9 P b B (b) Hence find the exact value of cos θ.

40 5. In the diagram AB = 15, BC = 6 and CF = 8 (a) Write down the coordinates of D and F (b) Calculate the size of angle DBF. z H E y G F D C A B x 6. The diagram shows three cuboids placed on top of each other. Two of the cuboids are equal in size 10 cm by cm by 5 cm. The third cuboid is centrally placed on the other two and has dimensions 6 cm by cm by 5 cm. z B 6 C y (a) Write down the coordinates of A, B and C. (b) Calculate the size of angle BAC A x

41 Vectors [SQA] 1. [SQA]. 1 4.Giventhatp = 5,q = 0 andr=,expressp q 1 rin component form. 1 A B C D hsn.uk.net Page 1 Questions marked [SQA] c SQA All others c Higher Still Notes

42 1 4.Giventhatu= 0 andv=,findu vincomponentform A B C D. 4 5 [SQA] 5. [SQA] 6. PQRSisaparallelogramwithverticesP(1,,),Q(4,, )andr(,1,1). Find the coordinates of S. [SQA] 7. hsn.uk.net Page Questions marked [SQA] c SQA All others c Higher Still Notes

43 8.Thevectoruhascomponents 0. 4 Whichofthefollowingisaunitvectorparalleltou? A. 5 i +4 5 k B. i +4k C. 7 i k D. 1 i +1 4 k 9.Ifu =k 1,wherek >0anduisaunitvector,determinethevalueofk. 0 A. 1 B. 1 8 C. 1 D [SQA] 10. Calculatethelengthofthevectori j + k. hsn.uk.net Page Questions marked [SQA] c SQA All others c Higher Still Notes

44 11.Thediagramshowsasquare-basedpyramidP,QRST. TS, TQand TPrepresent f, g and h respectively. P R h Q S f T g Express RPintermsof f,gandh. A. f +g h B. f g+h C. f g h D. f +g+h [SQA] 1. Thevectorsp,qandraredefinedasfollows: p =i j +k,q =4i j+k,r=4i j +k. (a)findp q+rintermsofi,jandk. 1 (b)findthevalueof p q+r. [SQA] 1. hsn.uk.net Page 4 Questions marked [SQA] c SQA All others c Higher Still Notes

45 [SQA] 14. [SQA] 15. [SQA] 16. ShowthatP(,,),Q(4,4,1)andR(5,5,0)arecollinearandfindtheratioin which Q divides PR. 4 [SQA] 17. Aisthepoint (, 5,6),Bis (6,,4)andCis (1,0,1).ShowthatA,BandC are collinear and determine the ratio in which B divides AC GiventhatthepointsS( 4,5,1),T( 16, 4,16)andU( 4, 10,6)are collinear, calculate the ratio in which T divides SU. A. : B. : C. :5 D. :5 [SQA] 19. Showthatthevectorsa =i +j kandb =i j+kareperpendicular. 1 0.Ifu = 1 andv= t are perpendicular, what is the value of t? t 1 A. B. C. D. 1 hsn.uk.net Page 5 Questions marked [SQA] c SQA All others c Higher Still Notes

46 1.Thevectorsxi +5j +7kand i +j kareperpendicular. Whatisthevalueofx? A. 0 B. 1 C. 4 D. 10 [SQA]. Forwhatvalueoftarethevectorsu = t andv= 10 perpendicular? t [SQA]. D,EandFhavecoordinates (10, 8, 15), (1,, )and (,0,1)respectively. (a) (i)showthatd,eandfarecollinear. (ii)findtheratioinwhichedividesdf. 4 (b)ghascoordinates (k,1,0). GiventhatDEisperpendiculartoGE,findthevalueofk. 4 [SQA] 4. hsn.uk.net Page 6 Questions marked [SQA] c SQA All others c Higher Still Notes

47 [SQA] 5. Acuboidmeasuring11cmby5cmby7cmisplacedcentrallyontopofanother cuboidmeasuring17cmby9cmby8cm. Coordinates axes are taken as shown. z A B C y x O (a)thepointahascoordinates (0,9,8)andChascoordinates (17,0,8). Write down the coordinates of B. 1 (b) Calculate the size of angle ABC. 6 [SQA] 6. hsn.uk.net Page 7 Questions marked [SQA] c SQA All others c Higher Still Notes

48 [SQA] 7. Vectors p, q and r are represented A B on the diagram shown where angle ADC =0. q r Itisalsogiventhat p =4and q =. 0 (a)evaluatep.(q +r)andr.(p q). D p C 6 (b)find q +r and p q. 4 [SQA] 8. hsn.uk.net Page 8 Questions marked [SQA] c SQA All others c Higher Still Notes

49 [SQA] 9. [SQA] 0. A box in the shape of a cuboid is designed with circles of different sizes on each face. The diagram shows three of the circles, where the origin represents oneofthecornersofthecuboid.the z centresofthecirclesarea(6,0,7), B(0,5,6)andC(4,5,0). B FindthesizeofangleABC. 7 A O y x C hsn.uk.net Page 9 Questions marked [SQA] c SQA All others c Higher Still Notes

50 [SQA] 1. [SQA]. hsn.uk.net Page 10 Questions marked [SQA] c SQA All others c Higher Still Notes

51 [SQA]. [SQA] 4. 5.Giventhata= 4 anda.(a +b) =7,whatisthevalueofa.b? 0 A. 7 5 B C. 6 D. 18 [END OF QUESTIONS] hsn.uk.net Page 11 Questions marked [SQA] c SQA All others c Higher Still Notes

52 Vectors Homework Expressions & Functions A is the point (-,, 4) and B is (-1,, ). Find (a) the components of vector AB (b) the length of AB 1 Show that P(,, ), Q(4, 4, 1) and R(5, 5, 0) are collinear and find the ratio in which Q divides PR. 4 Show that the vectors a = i + j k and b = i j + k are perpendicular. 4 The point Q divides the line joining P( 1, 1, 0) to R(5,, ) in the ratio : 1. Find the coordinates of Q St Andrew s Academy Maths Dept Higher E&F 1.4

St Andrew s Academy Mathematics Department Higher Mathematics

St Andrew s Academy Mathematics Department Higher Mathematics VECTORS hsn.uk.net Higher Mathematics Vectors Contents Vectors 1 1 Vectors and Scalars EF 1 Components EF 1 3 Magnitude EF 3 4 Equal Vectors