If P, Q, R are the mid-points of the sides AB, BC and CA respectively of a triangle ABC and a, p, q are the position vectors of A,P,Q respectively, then what is position vector of R?

Options

A. `2vec(a)-(vec(p)-vec(q))`
B. `(vec(p)-vec(q))-2vec(a)`
C. `vec(a)-(vec(p)-vec(q))`
D. `vec(a)//2-(vec(p)-vec(q))//2`

Answer

C

Explanation

Let the position vectors of B, C and R are `vec(b), vec(c) and vec(r)` respectively.

`:.vec(p)=(vec(a)+vec(b))/(2) rArr vec(b)=2vec(p)-vec(a)`
`vec(q)=(vec(b)+vec(c))/(2)`
`rArr vec(c)=2 vec(q)-vec(b)`
`rArr vec(c)=2vec(q)-(2vec(p)-vec(a))=2vec(q)-2vec(p)+vec(a)`
and `vec(r)=(vec(a)+vec(c))/(2)`
`=(2vec(q)-2vec(p)+vec(a)+vec(a))/(2)=vec(q)-vec(p)+vec(a)=vec(a)-(vec(p)-vec(q))`