Vectors Questions and Answers - Form 2 Topical Mathematics

Questions

1. If OA = 12i + 8j and OB = 16i + 4j. Find the coordinates of the point which divides AB
   internally in the ratio 1:3
2. Find scalars m and n such that
   m(⁴₃) + n(^-3₂) = (⁵₈)
3. In a triangle OAB, M and N are points on OA and OB respectively, such that OM: MA = 2:3 and ON: NB = 2:1. AN and BM intersect at X. Given that OA = a and OB = b.
   
   
   1. Express in terms of a and b
      1. BM
      2. AN
   2. By taking BX = t and AX = hAN, where t and h are scalars, express OX in two different ways
   3. Find the values of the scalars t and h
   4. Determine the ratios in which X divides :-
      1. BM
      2. AN
4. OABC is a parallelogram, M is the mid-point of OA and AX = ²/₇AC, OA=a and OC = c.
   
   
   1. Express the following in terms of a and c
      1. MA
      2. AB
      3. AC
      4. AX
   2. Using triangle MAX, express MX in terms of a and c
   3. The co-ordinates of A and B are (1, 6, 8) and (3, 0, 4) respectively. If O is the origin and P the midpoint of AB. Find;
      1. Length of OP
      2. How far are the midpoints of OA and OB?
5.  
   1. If A, B & C are the points (2, - 4), (4, 0) and (1, 6) respectively, use the vector method to find the coordinates of point D given that ABCD is a parallelogram.
   2. The position vectors of points P and Q are p and q respectively. R is another point with position vector r = ³/₂q - ½p. Express in terms of p and q
      1. PR
      2. PQ, hence show that P, Q & R are collinear.
      3. Determine the ratio PQ : QR
6. The figure shows a triangle of vectors in which OS: SP = 1:3, PR:RQ = 2:1 and T is the midpoint of OR.
   
   
   1. Given that OP = p and OQ = q, express the following vectors in terms of P and q
      1. OR
      2. QT
   2. Express TS in terms of p and q and hence show that the points Q, T and S are collinear.
   3. M is a point on OQ such that OM = KOQ and PTM is a straight line. Given that
      PT: TM = 5:1, find the value of k
7. Given that a = (^-3₂), b = (⁴_-6) and c = (⁵_-10) and that p = 3a – ½b +¹/₁₀c
   Express p as a column vector and hence calculate its magnitude /P/ correct to two decimal places
8. In a triangle OAB, M and N are points on OA and OB respectively, such that OM:MA= 2:3 and ON:NB= 2:1. AN and BM intersect at X. Given that OA = a and OB = b
   
   1. Express in terms of a and b:-
      1. BM
      2. AN
   2. Taking BX = kBM and AX =hAN where k and h are constants express OX in terms of
      1. a, b and k only
      2. a, b, and h only
   3. Use the expressions in (b) above to find values of k and h
9. In the figure below OAB is a triangle in which M divides OA in the ratio 2:3 and N divides OB in the ratio 4:1. AN and BM intersects at X.
   
   
   1. Given that OA = a and OB = b, express in terms of a and b
      1. AN
      2. BM
      3. AB
   2. If AX = sAN and BX = tBM, where s and t are constants, write two expressions for OX in terms of a, b, s and t. Find the value of s and t hence write OX in terms of a and b
10. Given that:- r = 5i – 2j and m = -2i + 6j – k are the position vectors for R and M respectively. Find the length of vector RM.
11. OABC is a trapezium in which OA = a and AB = b. AB is parallel to OC with 2AB = OC.
    T is a point on OC produced so that OC: CT = 2:1. AT and BC intersect at X so that BX = hBC and AX = KAT
    
    
    1. Express the following in terms of a and b:-
       1. OB
       2. BC
    2. Express CX in terms of a, b and h
    3. Express CX in terms of a, b and k
    4. Hence calculate the values of h and k
12. Given that a = 2i + j – 2k and b = -3i + 4j – k find :-
    | a + b|.
13. In the figure below, E is the mid-point of BC. AD:DC=3:2 and F is the meeting point of
    BD and AE. If AB = b and AC = c;
    
    
    1. Express BD and AE in terms of b and c
    2. If BF =tBD and AF =nAE, find the values of t and n
    3. State the ratios in which F divides BD and AE
14. The coordinates of point O, A, B and C are (0, 0) (3, 4) (11, 6) and (8, 2) respectively.
    A point P is such that the vector OP, BA, BC satisfy the vector equation OP = BA + ½ BC. Find the coordinates of P.
15. A point Q divides AB in the ratio 7:2. Given that A is (-3, 4) and B(2, -1).
    Find the co-ordinates of Q

Answers