RI Vectors C2B Add Prac (Qn)

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 _____________________________________________ Additional Practice Questions for Chapter 2B: Vectors II Page 1 of 4 Additional Practice Quest ions for Chapter 2B: Vectors II – Equation of Straight Lines 1 Referred to the origin O, the position vector of the point A is 226ij k and the cartesian equation of the line l is 12 6x yz  . Find (i) the position vector of the foot of the perpendicular from A to l, [3] (ii) the perpendicular distance from A to l, [2] 2 The line 1l has equation 12 12 , 3      r where  is a parameter. (i) Find the exact shortest distance between the origin O and 1.l [3] Another line 2l has equation 13 .229 x yz  (ii) Write down a vector equation of 2.l [1] (iii) Show that 1l and 2l are skew lines. [3] (iv) Find a vector that is perpendicular to both 1l and 2.l [1] The points P and Q lie on 1l and 2l respectively, such that the line PQ is perpendicular to both 1l and 2.l (v) Find a vector equation of the line PQ. [3] 3 The line 1l has equation 55 3 x z m   , 2y  and the line 2l has equation 5 x y , 0z  , where m is a constant. It is given that 1l and 2l intersect at point A. (i) Find the value of m, and the coordinates of A. [4] (ii) Find the position vector of the point P on 1l such that OP is perpendicular to 1l , where O is the origin. [3] (iii) Find a vector equation of the line which is a reflection of 2l in 1l . [3]

Raffles Institution H2 Mathematics 2025 Year 5 ____________________________________ _____ ______________________________________________ Additional Practice Questions for Chapter 2B: Vectors II Page 2 of 4 4 The points A and B have position vectors 8 3 2     and 2 3 4     respectively. (i) Show that 22 6AB . [1] (ii) Find the cartesian equation for the line AB . [2] (iii) The line l has equation r 22 36 45 t        . Find the length of the projection of AB onto l . [2] (iv) Calculate the acute angle between AB and l , giving your answer correct to the nearest degree. [2] (v) Find the position vector of the foot N of the perpendicular from A to l . Hence find the position vector of the image of A in the line l . [4] 5 With reference to the origin O, the points A and B have position vectors 2ai j k   and 25bj k respectively. (i) Find a vector equation of the line l1 that passes through point A and is parallel to the vector a. [ 1 ] (ii) Find the exact length of projection of b on l1. Hence find d, the exact perpendicular distance from the point B to l1. [4] (iii) Using the value of d found in part (ii), find the position vector of the point C, the foot of perpendicular from the point B to l1. [3] (iv) The line l2 passes through point B and is parallel to vector b. Find a cartesian equation of l3 which is the reflection of l2 in l1. [3] 6 The lines 1l and 2l have equations 61 31 01           r and 21 11 45               r respectively, where ,.   (i) Find the acute angle between 1l and 2.l [2] (ii) The points P and R lie on 1l and 2l respectively such that P is the reflection of the point R in the line l. The 3 lines intersect at the point (0,3, 6).Q  Find a possible pair of vectors QP  and QR  such that PQR is acute and 5.QP   Hence, or otherwise, find the vector equation of the line l. [4]

Raffles Institution H2 Mathematics 2025 Year 5 ____________________________________ _____ ______________________________________________ Additional Practice Questions for Chapter 2B: Vectors II Page 3 of 4 7 Relative to the origin O , the points ,A B and C have position vectors 25ij k , ik and 2 ij respectively. The line 1l passes through A and is parallel to the vector 3 2 ij . The line 2l passes through the points B and C . (i) Find a vector equation of the line 1l . [1] (ii) Show that the lines 1l and 2l intersect and that the position vector OX   of the point of intersection is 765ij k . [3] The vector abij k is perpendicular to both 1l and 2l . Find the values of a and b . Hence give an equation of the line 3l that is concurrent with 1l and 2l and is perpendicular to both 1l and 2l . If V is a point on 3l such that VABX is a tetrahedron with base ABX and height 145 units, find the coordinates of V . [7] 8 The figure shows a right pyramid VABCD with a square base ABCD, standing horizontally on a cuboid ABCDEFGH. It is given that VA = VB = VC = VD = 5 cm, EF = FG = 4 cm and AE = 2 cm, as shown in the diagram. O is the centre of the square base EFGH. Perpendicular unit vectors i, j, k are parallel to EF, FG, EA respectively. (i) Show that the height of the figure, OV is 21 7 . [2] (ii) State the geometrical meaning of 21 7 VA OV     . [1] (iii) Find the equation of the line passing through B and V. [2] 5 V E F G C B A D H O 2 4 i k j

Raffles Institution H2 Mathematics 2025 Year 5 ____________________________________ _____ ______________________________________________ Additional Practice Questions for Chapter 2B: Vectors II Page 4 of 4 9 Relative to an origin O , points A and B have position vectors 3 4 1     and 1 2 0     respectively. The line l has vector equation 61 3 0 at a        r , where t is a real parameter and a is a constant. The line m passes through the point A and is parallel to the line OB . (i) Find the position vector of the point P on m such that OP is perpendicular to m . [4] (ii) Show that the two lines l and m have no common point. [3] (ii) If the acute angle between the line l and the z axis is 60 , find the exact values of the constant a . [3] 10 Referred to an origin O, the position vectors of two points A and B are a and b respectively. A line l has vector equation given by r = 1 3 a2 b a , where   . The point N is the foot of perpendicular from A to l. It is given that 2, 1ab and a is perpendicular to b. Find the position vector of N in terms of a and b. [5] 11 The four points , , , A BCD have position vectors pi , qj k , k and ij respectively, where p and q are positive real numbers. Find vector equations for the lines AB and CD . (i) Given that the lines AB and CD intersect at a point R , express q in terms of p , and show that the two lines cannot be perpendicular. Show also that .ARC R ABC D (ii) Given that AB and CD are perpendicular, express q in terms of p , and show that the angle between AC and BD is equal to the angle between AD and BC .