Regent College. Maths Department. Core Mathematics 4. Vectors

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1 Regent College Maths Department Core Mathematics 4 Vectors Page 1

2 Vectors By the end of this unit you should be able to find: a unit vector in the direction of a. the distance between two points (x 1, y 1, z 1 ) and (x 2, y 2, z 2 ) by d 2 = (x 1 x 2 ) 2 + (y 1 y 2 ) 2 + (z 1 z 2 ) 2 the vector equation of a line in the form r = a + μb the point of intersection of two lines By the end of this unit you should also know that : If a and b are two vectors with an angle θ between them then: a. b = a b cosθ Where a.b is the scalar product of a and b. If a and b are two non zero vectors then if a.b = 0 then the two vectors are perpendicular. C4 Core Mathematics Vectors Page 2

3 C4 Core Mathematics Vectors Page 3

4 Vectors Notation The book and exam papers like writing vectors in the form a = 3i 4j + 7k. It is allowed, and sensible, to re-write vectors in column form i.e. a = 3i 4j + 7k = Definitions, adding and subtracting, etcetera A vector has both magnitude (length) and direction. If you always think of a vector as a translation you will not go far wrong. Directed line segments The vector is the vector from A to B, (or the translation which takes A to B). This is sometimes called the displacement vector from A to B. A B Vectors in co-ordinate form Vectors can also be thought of as column vectors, 7 thus in the diagram AB =. 3 Negative vectors is the 'opposite' of and so = Adding and subtracting vectors (i) Using a diagram 7. 3 Geometrically this can be done using a triangle (or a parallelogram): Adding: a b a A 7 a+b B 3 a+b b

5 The sum of two vectors is called the resultant of those vectors. Subtracting: b a b a (ii) Using coordinates a + c b d = a + c b + d and a c b d = a c b d. Parallel and non - parallel vectors Parallel vectors Two vectors are parallel if they have the same direction one is a multiple of the other. Example: Which two of the following vectors are parallel? 6 4 2,, Solution: Notice that = but 2 1 is not a multiple of 4 2 and so 6 3 is parallel to 4 2 and so cannot be parallel to the other two vectors. 4 Example: Find a vector of length 15 in the direction of Solution: a = has length a = a = = 5 3 and so the required vector of length 15 = 3 5 is 3a = =. 9

6 Non-parallel vectors If a and b are not parallel and if α a + β b = γ a + δ b, then αa γ a = δ b β b (α γ) a = (δ β) b but a and b are not parallel and one cannot be a multiple of the other (α γ) = 0 = (δ β) α = γ and δ = β. Example: If a and b are not parallel and if b + 2a + β b = α a + 3b 5a, find the values of α and β. Solution: Since a and b are not parallel, the coefficients of a and b must balance out 2 = α 5 α = 7 and 1 + β = 3 β = 2. Modulus of a vector and unit vectors Modulus The modulus of a vector is its magnitude or length. If 7 then the modulus of Or, if c = 3 5 then the modulus of c = c = c = Unit vectors A unit vector is one with length 1. Example: Find a unit vector in the direction of Solution: a = has length a = a = = 13, and so the required unit vector is a = =

7 Position vectors If A is the point ( 1, 4) then the position vector of A is the vector from the origin to A, 1 usually written as = a =. 4 y For two points A and B the position vectors are A b a = a and = b a B To find the vector go from A O B b giving = a + b = b a O x Ratios Example: A, B are the points (2, 1) and (4, 7). M lies on AB in the ratio 1 : 3. Find the coordinates of M. Solution : 2 6 B = M is (2 5, 2 5) O M 1 A 3

8 Proving geometrical theorems Example: In a triangle OBC let M and N be the midpoints of OB and OC. Prove that BC = 2 MN and that BC is parallel to MN. Solution: Write the vectors as b, and as c. Then = ½ = ½ b and = ½ = ½ c. O To find, go from M to O using ½ b and then from O to N using ½ c = ½ b + ½ c = ½ c ½ b B M b N c Also, to find, go from B to O using b and then from O to C using c C = b + c = c b. But = ½ b + ½ c = ½ (c b) = ½ BC BC is parallel to MN and BC is twice as long as MN. Example: P lies on OA in the ratio 2 : 1, and Q lies on OB in the ratio 2 : 1. Prove that PQ is parallel to AB and that PQ = 2 / 3 AB. A Solution: Let a =, and b = P 1 b a a, b and = 2 / 3 b 2 / 3 a = 2 / 3 (b a) O 2 2 Q 1 B = 2 / 3 PQ is parallel to AB and PQ = 2 / 3 AB.

9 Three dimensional vectors Length, modulus or magnitude of a vector The length, modulus or magnitude of the vector = a = a = a + a + a, 2 2 2` a sort of three dimensional Pythagoras. a1 a = 2 a 3 is Distance between two points To find the distance between A, (a 1, a 2, a 3 ) and B, (b 1, b 2, b 3 ) we need to find the length of the vector. b1 a1 b1 a1 = b a = b 2 a 2 b2 a = 2 b 3 a 3 b3 a = AB = ( b a ) + ( b a ) + ( b a ) Scalar product a. b = ab cos θ where a and b are the lengths of a and b and θ is the angle measured from a to b. b θ a Note that (i) a. a = aa cos 0 o = a 2 (ii) a. (b + c) = a. b + a. c. (iii) a. b = b. a since cosθ = cos ( θ) In co-ordinate form a1 b1 a. b =. = a1b1 + a2b2 a 2 b = ab cos θ 2 a1 b1 or a. b = a 2. b 2 = a1b1 + a2b2 + a3b3 = ab cos θ. a 3 b 3

10 Perpendicular vectors If a and b are perpendicular then θ = 90 o and cos θ = 0 thus a perpendicular to b a. b = 0 and a. b = 0 either a is perpendicular to b or a or b = 0. Example: Find the values of λ so that a = 3i 2λj + 2k and b = 2i + λj + 6k are perpendicular. Solution: Since a and b are perpendicular a. b = λ. = + = λ 0 6 2λ 12 0 λ 2 = 9 λ = ± 3. Example: Find a vector which is perpendicular to a, Solution: Let the vector c, p 1 q. 1 r 2 = 0 and 1 1, and b, 2 p q, be perpendicular to both a and b. r p 3 q. 1 r 1 p q + 2r = 0 and 3p + q + r = 0. = 0 Adding these equations gives 4p + 3r = Notice that there will never be a unique solution to these problems, so having eliminated one variable, q, we find p in terms of r, and then find q in terms of r. 3r 5r p = q = 4 4 c is any vector of the form 3r 4 5r, 4 r

11 and we choose a sensible value of r = 4 to give Angle between vectors Example: Find the angle between the vectors = 4i 5j + 2k and = i + 2j 3k, to the nearest degree. Solution: a = First re-write as column vectors (if you want) and b = a = a = = 45 = 3 5, b = b = = 14 and a. b = = = 20 a. b = ab cos θ 20 = cosθ 20 cos θ = = θ = 143 o to the nearest degree. Vector equation of a straight line x r = y is usually used as the position vector of a z general point, R. R l A In the diagram the line l passes through the point A and is parallel to the vector b. To go from O to R first go to A, using a, and then from A to R using some multiple of b. a r O b

12 The equation of a straight line through the point A and parallel to the vector b is r = a + λ b. Example: Find the vector equation of the line through the points M, (2, 1, 4), and N, ( 5, 3, 7). Solution: We are looking for the line through M (or N) which is parallel to the vector. = n m = equation is r = = λ Example: Show that the point P, ( 1, 7, 10), lies on the line r = Solution: λ The x co-ord of P is 1 and of the line is 1 λ 1 = 1 λ λ = 2. In the equation of the line this gives y = 7 and z = 10 P, ( 1, 7, 10) does lie on the line. Intersection of two lines 2 Dimensions Example: Find the intersection of the lines l 1, r = λ, and 2, μ 3 + = + 2 l r 3 1. Solution: We are looking for values of λ and μ which give the same x and y co-ordinates on each line. Equating x co-ords 2 λ = 1 + μ equating y co-ords 3 + 2λ = 3 μ

13 Adding 5 + λ = 4 λ = 1 μ = 2 lines intersect at (3, 1). 3 Dimensions This is similar to the method for 2 dimensions with one important difference you can not be certain whether the lines intersect without checking. You will always (or nearly always) be able to find values of λ and μ by equating x coordinates and y coordinates but the z coordinates might or might not be equal and must be checked. Example: Investigate whether the lines 2 1 l 1, r = + 1 λ 2 and l 2, r = 3 1 and if they do find their point of intersection μ 3 intersect 1 Solution: If the lines intersect we can find values of λ and μ to give the same x, y and z coordinates in each equation. Equating x coords 2 λ = 3 + μ, I equating y coords 1 + 2λ = 1 + 3μ, II equating z coords 3 + λ = 5 + μ. III 2 I + II 5 = 5 + 5μ μ = 2, in I λ = 3. We must now check to see if we get the same point for the values of λ and μ In l 1, λ = 3 gives the point ( 1, 7, 6); in l 2, μ = 2 gives the point ( 1, 7, 7). The x and y co-ords are equal (as expected!), but the z co-ordinates are different and so the lines do not intersect.

14 Example 1 The points A and B have coordinates (-3, -3, -3) and (0, 0, 2) respectively. The line l 1 which passes through A has equation 3 0 r = 3 + t Show that AB is perpendicular to l1 Exam papers have only recently started to use this type of notation for vector questions but the same principles apply. To find AB we simply need to subtract the two position vectors 3 AB = 3 5 The second bracket in the equation of the line is the direction component of the line. The direction component is used time and again in vectors questions. If the line is perpendicular to AB then the scalar product of the two should equal zero = = 0 as required. This is a very regularly asked question and it should be a good opportunity to pick up some marks. C4 Core Mathematics Vectors Page 4

15 Example 2 The points A and B have coordinates (2, 3, -3) and (1, -2, -2) respectively. The line l 1 which passes through A has equation 2 1 r = 3 + t The line l 2 which passes through B has equation 1 2 r = 2 + s Show that the lines l 1 and l 2 intersect, and find the coordinates of the point of intersection. Another standard question as all you have to do is equate the i,j,k components and solve the simultaneous equations. Therefore: 2 + t = 1 + 2s (1) 3 t = -2 3s -3 4t = s Adding equations (1) and gives: C4 Core Mathematics Vectors Page 5

16 5 = -1 s s = -6 hence t = -13 Substituting the value of t into l 1 gives: (-11, 16, 49) as the point of intersection. Check that substituting the value of s into l 2 gives the same answer. The next example involves more concepts but hopefully it is manageable. Example 3 Relative to a fixed point O, the points A and B have position vectors i + 2j -7k and 8i + 16j +7k respectively. a) Find the vector AB. b) Find the cosine of OAB c) Show that for all values of µ, the point P with position vector µi + 2µj + (2µ -9)k lies on the line through A and B. d) Find the value of µ for which OP is perpendicular to AB. e) Hence find the foot of the perpendicular from O to AB. a) Once again subtract the two position vectors. AB = 7i + 14j +14k C4 Core Mathematics Vectors Page 6

17 b) Find the cosine of OAB The use of a diagram may help here. OA AB OB We can use the scalar product to find the angle between two vectors BUT both of the vectors must be pointing towards the angle or away from it. Therefore in this case we will use AO Scalar Product a.b = a b cosθ Where a = AO and b = AB AO = -i - 2j +7k AB = 7i + 14j +14k a.b = = 63 a = 54 b = 441 Therefore cosθ= a.b ab C4 Core Mathematics Vectors Page 7

18 cosθ = cosθ = c) Show that for all values of µ, the point P with position vector µi + 2µj + (2µ -9)k lies on the line through A and B. When µ = 1 the point P has position vector i + 2j -7k (ie at A) When µ = 8 the point P has position vector 8i + 16j +7k (ie at B) Therefore for any other value of µ the point P will lie on the line through A and B d) Find the value of µ for which OP is perpendicular to AB The scalar product of OP and AB is equal to zero at the point where the two vectors are perpendicular. (7i + 14j +14k). (µi + 2µj + (2µ -9)k) = 0 7µ + 28µ + 28µ = 0 µ = 2 e) Hence find the foot of the perpendicular from O to AB. We have just shown in part (d) that when µ = 2 OP is perpendicular to AB and that P is on the line between A and B. Therefore we only need to substitute µ = 2 into the position vector for P to find the foot of the perpendicular. C4 Core Mathematics Vectors Page 8

19 µi + 2µj + (2µ -9)k 2i + 4j -5k Example 4 The line l 1 has equation 7 1 r = 10 + t where t is a parameter. The point A has coordinates (2, 5, a), where a is a constant. The point B has coordinates (b, 12, 12), where b is a constant. The points A and B lie on the line l 1. a) Find the values of a and b. Given that the point O is the origin, and that the point P lies on l 1 such that OP is perpendicular to l 1, b) find the coordinates of P. c) Hence find the distance OP, giving your answer in surd form. a) Find the values of a and b The x coord of A is 2 therefore: 7 + t = 2 t = -5 So the z coord must be: C4 Core Mathematics Vectors Page 9

20 = 19 a = 19 The z coord of B is 12 therefore: 14 - t = 12 t = 2 So the x coord must be: = 9 b = 9 b) find the coordinates of P The scalar product of OP and the direction component of the line l 1 will equal zero if the two are perpendicular. Therefore: OP = (7 + t, 10 + t, 14 t) Direction component = (1, 1, -1) (7 + t, 10 + t, 14 t). (1, 1, -1) = t t 14 + t = 0 t = -1 Substituting this value of t into the equation of the line gives the coordinates of P as: (6, 9, 15) C4 Core Mathematics Vectors Page 10

21 c) Hence find the distance OP, giving your answer in surd form. Using Pythagoras in three dimensions: OP = ( ) = 342 C4 Core Mathematics Vectors Page 11

22 Past paper questions vectors 1. The line l 1 has vector equation and the line l 2 has vector equation where λ and µ are parameters. 3 1 r = 1 + λ r = 4 + µ 1, 2 0 The lines l 1 and l 2 intersect at the point B and the acute angle between l 1 and l 2 is θ. (a) Find the coordinates of B. (b) Find the value of cos θ, giving your answer as a simplified fraction. (4) (4) The point A, which lies on l 1, has position vector a = 3i + j + 2k. The point C, which lies on l 2, has position vector c = 5i j 2k. The point D is such that ABCD is a parallelogram. (c) Show that AB = BC. (d) Find the position vector of the point D. 2. The line l 1 has vector equation (C4, June 2005 Q7) where λ is a parameter. r = 8i + 12j + 14k + λ(i + j k), The point A has coordinates (4, 8, a), where a is a constant. The point B has coordinates (b, 13, 13), where b is a constant. Points A and B lie on the line l 1. (a) Find the values of a and b. Given that the point O is the origin, and that the point P lies on l 1 such that OP is perpendicular to l 1, C4 Core Mathematics Vectors Page 12

23 (b) find the coordinates of P. (b) Hence find the distance OP, giving your answer as a simplified surd. (5) (C4, Jan 2006 Q6) 3. The point A, with coordinates (0, a, b) lies on the line l 1, which has equation r = 6i + 19j k + λ(i + 4j 2k). (a) Find the values of a and b. The point P lies on l 1 and is such that OP is perpendicular to l 1, where O is the origin. (b) Find the position vector of point P. (6) Given that B has coordinates (5, 15, 1), (c) show that the points A, P and B are collinear and find the ratio AP : PB. (4) (C4, June 2006 Q5) 4. The point A has position vector a = 2i + 2j + k and the point B has position vector b = i + j 4k, relative to an origin O. (a) Find the position vector of the point C, with position vector c, given by c = a + b. (b) Show that OACB is a rectangle, and find its exact area. The diagonals of the rectangle, AB and OC, meet at the point D. (1) (6) (c) Write down the position vector of the point D. (d) Find the size of the angle ADC. (1) (6) (C4, Jan 2007Q7) C4 Core Mathematics Vectors Page 13

24 1 5. The line l 1 has equation r = λ The line l 2 has equation r = 3 + µ (a) Show that l 1 and l 2 do not meet. (4) The point A is on l 1 where λ = 1, and the point B is on l 2 where µ = 2. (b) Find the cosine of the acute angle between AB and l 1. (6) (C4, June 2007 Q5) 6. The points A and B have position vectors 2i + 6j k and 3i + 4j + k respectively. The line l 1 passes through the points A and B. (a) Find the vector AB. (b) Find a vector equation for the line l 1. A second line l 2 passes through the origin and is parallel to the vector i + k. The line l 1 meets the line l 2 at the point C. (c) Find the acute angle between l 1 and l 2. (d) Find the position vector of the point C. (4) (C4, Jan 2008 Q6) C4 Core Mathematics Vectors Page 14

25 7. With respect to a fixed origin O, the lines l 1 and l 2 are given by the equations l 1 : r = ( 9i + 10k) + λ(2i + j k) l 2 : r = (3i + j + 17k) + μ(3i j + 5k) where λ and μ are scalar parameters. (a) Show that l 1 and l 2 meet and find the position vector of their point of intersection. (b) Show that l 1 and l 2 are perpendicular to each other. (6) The point A has position vector 5i + 7j + 3k. (c) Show that A lies on l 1. (1) The point B is the image of A after reflection in the line l 2. (d) Find the position vector of B. (C4, June 2008 Q6) 8. With respect to a fixed origin O the lines l 1 and l 2 are given by the equations l 1 : r = λ l 2 : r = 11 p + μ q 2 2 where λ and μ are parameters and p and q are constants. Given that l 1 and l 2 are perpendicular, (a) show that q = 3. Given further that l 1 and l 2 intersect, find (b) the value of p, (6) (c) the coordinates of the point of intersection. C4 Core Mathematics Vectors Page 15

26 9 The point A lies on l 1 and has position vector 3. The point C lies on l Given that a circle, with centre C, cuts the line l 1 at the points A and B, (d) find the position vector of B. (C4, Jan2009 Q4) 9. Relative to a fixed origin O, the point A has position vector (8i + 13j 2k), the point B has position vector (10i + 14j 4k), and the point C has position vector (9i + 9j + 6k). The line l passes through the points A and B. (a) Find a vector equation for the line l. (b) Find CB. (c) Find the size of the acute angle between the line segment CB and the line l, giving your answer in degrees to 1 decimal place. (d) Find the shortest distance from the point C to the line l. The point X lies on l. Given that the vector CX is perpendicular to l, (e) find the area of the triangle CXB, giving your answer to 3 significant figures. (C4, June 2009 Q7) 10. The line l 1 has vector equation and the line l 2 has vector equation where λ and μ are parameters. 6 r = 4 + λ r = 4 + µ The lines l 1 and l 2 intersect at the point A and the acute angle between l 1 and l 2 is θ. C4 Core Mathematics Vectors Page 16

27 (a) Write down the coordinates of A. (b) Find the value of cos θ. (1) The point X lies on l 1 where λ = 4. (c) Find the coordinates of X. (d) Find the vector AX. (e) Hence, or otherwise, show that AX = (1) The point Y lies on l 2. Given that the vector YX is perpendicular to l 1, (f) find the length of AY, giving your answer to 3 significant figures. (C4, Jan 2010 Q4) The line l 1 has equation r = 3 + λ 2, where λ is a scalar parameter The line l 2 has equation r = 9 + µ 0, where µ is a scalar parameter. 3 2 Given that l 1 and l 2 meet at the point C, find (a) the coordinates of C. The point A is the point on l 1 where λ = 0 and the point B is the point on l 2 where μ = 1. (b) Find the size of the angle ACB. Give your answer in degrees to 2 decimal places. (c) Hence, or otherwise, find the area of the triangle ABC. (4) (5) (C4, June 2010 Q7) C4 Core Mathematics Vectors Page 17

28 12. Relative to a fixed origin O, the point A has position vector i 3j + 2k and the point B has position vector 2i + 2j k. The points A and B lie on a straight line l. (a) Find AB. (b) Find a vector equation of l. The point C has position vector 2i + pj 4k with respect to O, where p is a constant. Given that AC is perpendicular to l, find (c) the value of p, (d) the distance AC. 13. With respect to a fixed origin O, the lines l 1 and l 2 are given by the equations (4) (C4, Jan2011 Q4) l 1 : r = λ 2, l 2 : r = μ 2 3, 1 where μ and λ are scalar parameters. (a) Show that l 1 and l 2 meet and find the position vector of their point of intersection A. (b) Find, to the nearest 0.1, the acute angle between l 1 and l 2. (6) The point B has position vector (c) Show that B lies on l 1. (d) Find the shortest distance from B to the line l 2, giving your answer to 3 significant figures. (4) (C4, June 2011 Q6) (1) C4 Core Mathematics Vectors Page 18

29 14. Relative to a fixed origin O, the point A has position vector (2i j + 5k), the point B has position vector (5i + 2j + 10k), and the point D has position vector ( i + j + 4k). The line l passes through the points A and B. (a) Find the vector AB. (b) Find a vector equation for the line l. (c) Show that the size of the angle BAD is 109, to the nearest degree. (4) The points A, B and D, together with a point C, are the vertices of the parallelogram ABCD, where AB = DC. (d) Find the position vector of C. (e) Find the area of the parallelogram ABCD, giving your answer to 3 significant figures. (f) Find the shortest distance from the point D to the line l, giving your answer to 3 significant figures. (C4, Jan2012 Q7) 15. Relative to a fixed origin O, the point A has position vector (10i + 2j + 3k), and the point B has position vector (8i + 3j + 4k). The line l passes through the points A and B. (a) Find the vector AB. (b) Find a vector equation for the line l. The point C has position vector (3i + 12j + 3k). The point P lies on l. Given that the vector CP is perpendicular to l, (c) find the position vector of the point P. (6) (C4, June 2012 Q8) C4 Core Mathematics Vectors Page 19

30 16. With respect to a fixed origin O, the lines l 1 and l 2 are given by the equations where λ and µ are scalar parameters. l 1 : r = (9i + 13j 3k) + λ (i + 4j 2k) l 2 : r = (2i j + k) + µ (2i + j + k) (a) Given that l 1 and l 2 meet, find the position vector of their point of intersection. (b) Find the acute angle between l 1 and l 2, giving your answer in degrees to 1 decimal place. Given that the point A has position vector 4i + 16j 3k and that the point P lies on l 1 such that AP is perpendicular to l 1, (5) (c) find the exact coordinates of P. (5) (C4, Jan2013 Q7) 17. With respect to a fixed origin O, the line l has equation 13 2 r = 8 + λ 2, where λ is a scalar parameter. 1 1 The point A lies on l and has coordinates (3, 2, 6). The point P has position vector ( pi + 2pk) relative to O, where p is a constant. Given that vector PA is perpendicular to l, (a) find the value of p. (4) Given also that B is a point on l such that <BPA = 45, (b) find the coordinates of the two possible positions of B. (5) (C4, June 2013 Q8) C4 Core Mathematics Vectors Page 20

31 18. Relative to a fixed origin O, the point A has position vector 21i 17j + 6k and the point B has position vector 25i 14j + 18k. The line l has vector equation a 6 r = b + λ c 10 1 where a, b and c are constants and λ is a parameter. Given that the point A lies on the line l, (a) find the value of a. Given also that the vector AB is perpendicular to l, (b) find the values of b and c, (c) find the distance AB. (5) The image of the point B after reflection in the line l is the point B. (d) Find the position vector of the point B. (C4, June 2013_R Q6) 19. Relative to a fixed origin O, the point A has position vector 1 and the point B has position vector The line l 1 passes through the points A and B. (a) Find the vector AB. (b) Hence find a vector equation for the line l 1. (1) C4 Core Mathematics Vectors Page 21

32 0 The point P has position vector 2. 3 Given that angle PBA is θ, (c) show that 1 cosθ =. 3 The line l 2 passes through the point P and is parallel to the line l 1. (d) Find a vector equation for the line l 2. The points C and D both lie on the line l 2. Given that AB = PC = DP and the x coordinate of C is positive, (e) find the coordinates of C and the coordinates of D. (f) find the exact area of the trapezium ABCD, giving your answer as a simplified surd. (4) (C4, June 2014 Q8) 20. With respect to a fixed origin, the point A with position vector i + 2j + 3k lies on the line l 1 with equation 1 0 r = 2 + λ 2, where λ is a scalar parameter, 3 1 and the point B with position vector 4i + pj + 3k, where p is a constant, lies on the line l 2 with equation 7 3 r = 0 + µ 5, where μ is a scalar parameter. 7 4 (a) Find the value of the constant p. (1) C4 Core Mathematics Vectors Page 22

33 (b) Show that l 1 and l 2 intersect and find the position vector of their point of intersection, C. (c) Find the size of the angle ACB, giving your answer in degrees to 3 significant figures. (d) Find the area of the triangle ABC, giving your answer to 3 significant figures. (4) (C4, June 2014_R Q6) C4 Core Mathematics Vectors Page 23