RVHS H2 Math Vectors Qns

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Vectors 1 Vectors 1. AJC/I/13 The points P and Q have position vectors ij− and 3 13 6i j k++ respectively. The plane 1 contains the point P and the line 1 , 02 x zy=− − = . (i) Find a vector equation of the plane 1 in scalar product form. [3] (ii) Find the position vector of the foot of the perpendicular from Q to 1 . [2] The line l1 passes through the points P and Q. (iii) The line l2 is the reflection of the line l1 about the plane 1 . Find a vector equation of l2. [3] The plane 2 has the equation 6 4 a b  •= r . Find the values of a and b such that (iv) 1 and 2 are parallel and at a distance of 224 apart. [3] (v) 1 and 2 are intersecting. [1] 2. AJC/II/2 The diagram shows a roof, with horizontal rectangular base OBCD, where OB = 10m and BC = 6m. The triangular planes ODE and BCF are vertical and the ridge EF is horizontal to the base. The planes OBFE and DCFE are each inclined at an angle  to the horizontal, where 4tan 3 = . The point O is taken as origin and vectors i, j, k, each of length 1m, are taken along OB, OD and vertically upwards from O respectively. (i) Show that OF  = 10i + 3j + 4k. [2] (ii) Find the acute angle between the line OF and the plane BDE, giving your answer to the nearest 0.1  . [4] 3. ACJC/2016/I/111 O B C D E F i j k

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Vectors 2 Referred to the origin O, the position vectors of the points A, B and C are a, b and c respectively. Given that 4a b a c =  , where 4bc and a is a non-zero vector, (i) show that 4b c a −= where  is a scalar. [1] (ii) Hence evaluate bc , given that the area of triangle OAB is 126 and 3= . [2] (iii) Give the geometrical meaning of bc . [1] It is also given that b is a unit vector, 5a = , 2c = and 43b c a−= . (iv) By considering ( ) ( )44b c b c−  − , find the angle between b and c . [3] 4. ASRJC/Prelim/2021/II/3 Three distinct vectors ,ab and c are each of unit length such that   + + =a b c 0 , where ,,   are non-zero scalars. (i) Show that ( ) ( )  = a b c a and ( ) ( )  = b a c b . [3] (ii) By considering the sines of angles between and bc , and ca , and and ab , show that sin sin sinCOA BOC AOB    ==   . [4] 5. VJC Prelim/2020/01/Q7 With reference to the origin O, the points A, B, P, Q and R have position vectors a , b , 2−ab , 23−−ab and 2 +ab respectively, where a and b are non-parallel vectors and 0ab  . Given that a is a unit vector and the area of triangle PQR is equal to the magnitude of b, show that 1sin 4 = , where θ is the angle between a and b. Hence, find the value of  . [5] M lies on PR such that 1 2PM MR= . Given that PR is perpendicular to OM, find the magnitude of b, giving your answer correct to 3 decimal places. [5] 6. CJC/2016/I/2 It is given that 2=ab . (i) Find p. [2] (ii) Give a geometrical interpretation of 1 bab . [1] (iii) Using the value of p found in part (i), find the exact value of 1 bab . [2] 7. MJC/I/5 The vectors a and b are given by 4 6 8 p= + −a i j k and 2 3 4 p= − +b i j k , where 0p .

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Vectors 3 With respect to an origin O, the points A and B have position vectors a and b respectively, where a and b are not parallel. (i) It is given that B lies on the line segment AC, such that 2BC k=− ba  . State the value of the constant k. Hence, find OC  in terms of a and b. [2] (ii) ON  is the projection vector of OB  onto OA  , where 3 4ON = a  and 4=a.b . Show that the length of OA is 43 3 . [2] (iii) Given also that b 2 2 2  =− , find the angle between OA  and OB  . [2] (iv) Given that OADC is a parallelogram and E lies on line segment AC such that : 2 : 3AE AC = , find DE  . [3] 8. CJC/9758/2025/I/8 The plane p passes through the points A, B and C with coordinates ( )1,0, 2 , ( )2, 1,3− and ( )4, 1,0−− respectively. (a) Show that a cartesian equation of p is 23x y z− − =− . [3] The line l has equation 22 0 1 , 51         = + −        −    r  . (b) Find the acute angle between l and p. [2] It is given that a variable point R lies on p and is at a distance of 22 from the point Q with coordinates ( )3, 4, 2− . (c) Find the foot of perpendicular from Q to p. [4] (d) Hence describe geometrically the path traced by R. [2] 9. PJC/II/4 The point A has position vector i + j + 3k and the plane,  , has equation ( )3 4 40.+=r i k  Find (i) the vector equation of the line l passing through A and normal to the plane,  , [1] (ii) the position vector of the foot of perpendicular from the point A to the plane,  . [3] A sphere of radius 2 units has its centre at A. Show that the shortest distance between the sphere and the plane,  , is 3 units. Hence find the position vector of the nearest point on the sphere to the plane. [5]

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Vectors 4 10. CJC/2010/II/2 The equations of the line l1 and the plane p1 are respectively,          − +           = 1 1 1 0 0 1 r and           −+           = 1 1 2 1 1 1 tsr , where , s, t   (i) Find the acute angle between l1 and p1. [3] (ii) A second plane p2 has equation 11 =              r . Given that the two planes p1 and p2 intersect at the line l2: 2 5;2 154 ==− − zyx , find the values of α and β. [3] (iii) The plane p3 with equation 12 =++ zbyx is parallel to l2. Find the value of b. Hence find the distance between l2 and p3. [4] 11. TJC Prelim 9758/2024/02/Q4 The line 1l passes through the point A with coordinates ( )1,0,4 and is perpendicular to the plane 1 with equation 2 4 3x y z− + =− . (a) Find the coordinates of the point B where 1l meets 1 . [4] (b) Verify that the point C with coordinates ( )5,9, 1− lies on 1 . [1] (c) Find a vector equation of the line 2l which is a reflection of the line AC in 1 . [3] (d) Find a vector equation, in scalar product form, of the plane 2 which contains 1l and 2l . [3] 12. MJC/2010/II/2 Three non -zero and non -parallel vectors p, q and r are such that 3 = p q p r . Show that 3 −=q r p , where  is a scalar. [2] It is also given that p is a unit vector, 5q = , 2r = and the angle between q and r is 1 5cos 6 − . By considering ( ) ( )33−  −q r q r , find the exact values of  . [4]

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Vectors 5 13. PJC/2010/II/3 The diagram above shows a partial design of the roof of the gallery of an amphitheatre. ABCD is an inclined rectangular roof, where AB = 10 m, and 5 mBC= . EFGH is a rectangle on a horizontal ground, where EF = 10 m and 4 mEH = . The points A and B are 3 m directly above E and F respectively. The points C and D are 6 m directly above G and H respectively. Point O is the centre of EFGH and is taken as the origin. Perpendicular unit vectors i, j, k are in the direction of EF  , EH  and EA  respectively. (i) Show that 10 4 3 AC  =  and find a vector equa