Vector question?

Question

Given that vector OA = (p, 4), OB = (4, 1) and OC = (1, 3),

(i) find the value of p if vector OA and BC are parallel.

Ans: -6

Please show me your workings clearly.....and explain too..Thanks

Answer 1

BC= OC-OB = (-3,2)
p/-3= 4/2 so p=-6 ( If parallel the components are proportional)

Answer 2

First, we must find vector BC. Vector BC is a vector that points from B to C, so:

BC = OC - OB = (1,3) - (4,1) = (-3, 2)

THE POINT: two vectors are parallel if we can get one vector from the other by multiplying it with some constant (it means that two vectors point in the same direction, only their magnitude is different).

So:

BC = k (OA) (where k is some constant above-mentioned)

(-3,2) = k (p,4)

we get two equations: -3 = pk ...(1)
2 = 4k...(2)

from equation (2) we get k =1/2, so substitute it back to equation (1) gives p = -6

Answer 3

Vector BC = (1 3) - (4 1) = (- 3 2)
Vector OA = (p 4)
OA = 2 BC
p = 2 x - 3
p = - 6