Vectors Higher Mathematics Supplementary Resources. Section A

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1 Education Resources Vectors Higher Mathematics Supplementar Resources Section This section is designed to provide eamples which develop routine skills necessar for completion of this section. R I have revised National 5 vectors and D coordinate.. If vector a = ( ) and vector b = ( ), find the resultant vector of: 4 a + b (b) a b a + b (d) a b (e) 5a b (f) a + 4b. If vector a = ( ) and vector b = ( 4), find the resultant vector of a + b (b) a b a + b (d) 5a b (e) a b (f) a + 4b. If vector p = ( 4 ) and vector q = ( ), find the resultant vector of p + q (b) p q p + q (d) p q (e) p 5q (f) 4p + q

2 4. If p = ( ) and q = ( ), find: p (b) q p + q (d) p q (e) p q (f) p + q 5. Three vectors are defined as B = ( ), CD = ( ) and EF = ( ), find: 5 B (b) CD EF 6. Three points, B and C have the coordinates (, 5, ), (,, ) and (, 4, ) respectivel. Find the vectors (b) B C (d) B (e) BC (f) C 7. The diagram shows the cuboid BCDEFG. is the origin and, C and D are aligned with the, and aes respectivel. The point F has coordinates (5,, 4). List the coordinates of the other si vertices. D C G E F (5,, 4) B 8. The diagram shows the square based pramid DBC. is the origin with and C aligned with the and aes respectivel. The point D has coordinates (6, 6, ). Write down the coordinates of the points, B and C. C D(6, 6,) B

3 R I can epress and manipulate vectors in the form ai + bj + ck.. Write the following vectors, given in unit vector form, in component form. a = i + j + k (b) b = i + j + k c = 4i + j (d) d = i + j k (e) e = i 6j 4k (f) f = i + k. Write the following vectors, given in component form, in unit vector form. 6 p = ( ) (b) q = ( ) r = ( 4) 7 9 (d) s = ( ) (e) t = ( ) (f) u = ( 5 ). Two vectors are defined, in unit vector form, as a = i + j + k and b = i j + 5k. Epress a + b in unit vector form. (b) Epress a b in unit vector form. Find a + b. (d) Find a b. 4. Two vectors are defined, in unit vector form, as p = i k and q = i j + k. Epress p + q in unit vector form. (b) Epress p 4q in unit vector form. Find p + q. (d) Find p 4q.

4 R I can calculate the scalar product and know that perpendicular vectors have a scalar product of ero.. Find the scalar product of each of the pairs of vectors below and state clearl which pairs are perpendicular. B = ( ) and CD = ( ). 5 6 (b) p = ( ) and q = ( ). a = i 4j + k and b = i + j + k (d) PQ = ( ) and RS = ( ). 8 (e) s = ( ) and t = ( ). (f) a = i + j + k and b = 5i + j + 4k. If p = and q = 7 and p and q are inclined at an angle of, find the scalar product p q.. If B = and C = 4 and B and C are inclined at an angle of 6, find the scalar product B C. 4. If a = and b = and p and q are inclined at an angle of 45, 4 find the scalar product p q.

5 Section B This section is designed to provide eamples which develop Course ssessment level skills NR I can determine whether or not coordinates are collinear, using the appropriate language, and can appl m knowledge of vectors to divide lines in a given ratio.. The point Q divides the line joining P(,, ) and R(5,, ) in the ratio 5:. Find the coordinates of Q.. John is producing a D design on his computer. Relative to suitable aes points in his design have coordinates P(, 4, 7), Q(, 8, ) and R(,, ). (b) Show that P, Q and R are collinear. Find the coordinates of S such that PS = 4PQ.. and B are the points (,, ) and (,, ) respectivel. B and C are the points of trisection of D, that is B = BC = CD. Find the coordinates of D. B C D 4. The points V, W and X are shown on the line opposite. V, W and X are collinear points such that WX = VW. Find the coordinates of X. V (,, ) W (,, ) X

6 5. QRS is a pramid. Q is the point (6,, ), R is (6, 8, ) and is (8, 4, ). T divides R in the ratio :. Find the coordinates of the point T. T (b) Epress QT in component form. S R Q 6. PQRST is a pramid with a rectangular base PQRS. V divides QR in the ratio : and TP = 7i j k, T PQ = 6i + 6j 6k and PS = 8i 4j + 4k S R Find TV in component form. P Q V

7 NR I can appl knowledge of vectors to find an angle in three dimensions.. surveor is checking a room for movement in the walls due to subsidence. She sets up points. Two of the points, and M, are on two different walls which meet perpendicularl along the ais, relative to the aes shown. The other point T is on the floor. The three points have coordinates (6,, 7), (, 5, 6) and (4, 5, ). Match the three points to the correct coordinates. M (b) Write T and TM in component form. Find the sie of angle TM. T. V, W and X have coordinates (,, ), (,, ) and (,, ) respectivel. Find VW and VX in component form. (b) Hence find the sie of angle WVX.. Three planes, Tango (T), Delta (D) and Bravo (B) are being tracked b radar. Relative to a suitable origin, the positions of the three planes are T(,, 8), D(,, 9) and B(8, 5, 7) Epress the vectors BT and BD in component form. (b) Find the sie of angle TBD.

8 4. cuboid measuring cm b 6cm b 6cm is placed centrall on top of another cuboid measuring 8cm b cm b 9cm. Coordinate aes are taken as shown. 6 The point has coordinates (,, 9) and the point C has coordinates (8,, 9). Write down the coordinates of B. (b) Find the sie of angle BC. B 8 6 C 9 5. square-based pramid, BCD, has a height of units and the square base has a length of 8 units. The coordinates of two points, and D are shown on the diagram. Write down the coordinates of the point B. D(4, 4,) (b) Determine the components of the vectors D and DB. Find the sie of angle DB. C B M is the midpoint of. M (8,,) (d) Write down the coordinates of C and M. (e) Determine the components of the vectors DC and DM. (f) Find the sie of angle CDM.

9 6. The diagram shows a cuboid BCDEFG with the lines, C and D ling on the aes. The point F has coordinates (8, 6, ), M is the midpoint of CG and N divides BF in the ratio :. State the coordinates of, M and N. (b) Determine the components of the vectors M and MN. Find the sie of angle MN. D C G M E B F (8, 6, ) N 7. The diagram shows a cube BCDEFG. B has coordinates (,, ) M is the centre of EFB and N is the centre of face GFBC. (b) Write down the coordinates of G. Find m and n, the position vectors of M and N. D G C N E M F B (,, ) Find the sie of angle MN. 8. In the diagram PQRSTUV is a cuboid. M is the midpoint of VR and N is the point on UQ such that UN = UQ. State the coordinates of T, M and N. (b) Determine the components of the vectors TM and TN. Find the sie of angle MTN. S R V M T P (6,, ) U (6,, ) N Q (6,, )

10 NR I know the properties of the scalar product and their uses.. Vectors p and q are defined b p = i k and q = 8i + 7j k. Determine whether or not p and q are perpendicular to each other. p. For what value of p are the vectors a = ( ) and b = ( 4) perpendicular? p. and B have coordinates (9, 7, 4), (,, ) respectivel. C has coordinates (k,, ). Given that B is perpendicular to CB, find the value of k. 4. The diagram shows vectors p and q. If p =, q = 4 and p. (p + q) = 5, find the sie of the acute angle θ between p and q. θ p q 5. The diagram shows vectors a, b and c. c 45 a b a =, b = and the angle between a and b is 45. c is perpendicular to a and to b. Evaluate the scalar product a. (a + b + c). a c b

11 6. The vectors a, b and c form an equilateral triangle of length units. Find the scalar product a. (b + c). (b) What does this tells us about the vectors a and b + c. 7. The vectors a, b and c are shown on the diagram. ngle PQR = 6. P S b c Q 6 a R It is also given that a = and b =. Evaluate a. (b + c) and c. (a b). (b) Find b + c and a b.

12 NR4 I have eperience of cross topic eam standard questions. Vectors and Polnomials. p and q are vectors given b p = ( k k ) and q = ( k ), where k >. k + p θ q If p q = k, show that k + k k =. (b) Show that (k + ) is a factor of k + k k and hence factorise full. Deduce the onl possible value of k. Vectors and Quadratics. P is the point (,, ), Q(,, ) and R(k,, ) Epress QP and QR in component form. (b) Show that cos PQ R = (k k+6) If angle PQR =, find the possible values of k.

13 nswers R. ( 5 ) (b) ( 5 ) (9 ) (d) ( ) (e) ( ) (f) (6 7 8 ) 5 5. ( 4) (b) ( 4) ( ) (d) ( 4) (e) ( 8) (f) ( 6) ( 6) (b) ( ) ( 8 ) (d) ( 6 ) (e) ( ) (f) ( ) (b) 6 (d) (e) (f) (b) 7 6. ( 5) (b) ( ) ( 4) (d) ( ) (e) ( ) (f) ( ) 7. (5,, ), B(5,, ), C(,, ), D(,, 4), E(5,, 4), G(,, 4) 8. (,, ), B(,, ), C(,, ) R 4. ( ) (b) ( ) ( ) (d) ( ) (e) ( 6) (f) ( ) 4. i + j + k (b) 6i j + 7k i 4j (d) 9i k (e) i + j k (f) 5j k. 5i + j + 6k (b) i + 7j k 65 (d) i 4j + 5k (b) 5i + 8j 5k 66 (d) 4 R. (b) (perpendicular) (d) (e) (f) (perpendicular) 4. 4

14 NR. Q(4,, ). QR = ( ), and PQ = ( 4 ) = ( ) with conclusion 4 (b) S(5,, 9). D(9, 4, ) 4. X(7, 7, 8) 5. T(4, 7, ) (b) QT = ( 7 ) 6. TV = ( 8 ) 6 NR 4. T(4, 5, ), (6,, 7), M(, 5, 6) (b) T = ( 5) and TM = ( ) VW = ( ) and VX = ( ) (b) BT = ( 5) and BD = ( 5 ) (b) B(,, 5) (b) B(8, 8, ) (b) D = ( (d) C(, 8, ),M(4,, ) 4 4 ) and DB = ( 4 4 ) (e) DC = ( 4 ) and DM = ( 4 ) (f) (8,, ), M(, 6, 5), N(8, 6, 4) (b) M = ( 6) and MN = ( ) G(,,) (b) m = ( 5 ) and n = ( ). 5 5

15 8. T(6,, ), M(,, 5), N(6,, ) (b) TM = ( 6 5 ) and TN = ( ) NR p q = therefore p and q are perpendicular.. p = 4. k = θ = a. (b + c) = (b) a is perpendicular to b + c 7. a. (b + c) =, c. (a b) = (b) b + c =, a b = 7. NR4 Vectors and Polnomials. Proof (b) (k + )(k + )(k ) k = as k > Vectors and Quadratics k. QP = ( ) and QR = ( ) (b) Proof k =,