If ` vec r_1, vec r_2, vec r_3` are the position vectors off thee collinear points and scalar `pa n dq` exist such that ` vec r_3=p vec r_1+q vec r_2,` then show that `p+q=1.`

Options

A. 0
B. 1
C. -1
D. 2

Answer

B

Explanation

Since `vec(r)_(1), vec(r)_(2) and vec(r)_(3)` are the position vector of three collinear points. Thus `vec(r)_(3)` is the position vector of the point which divides the joining of points whose position vectors are `vec(r)_(1) andvec(r)_(2)` in the ratio m:n.
So, `vec(r)_(3)=(mvec(r)_(1)+n vec(r)_(2))/(m+n)`
But as given, `vec(r)_(3)=mvec(r)_(1)+n vec(r)_(2)`
So, `(vec(mr_(1))+vec(n r_(2)))/(m+n)=mr_(1)+nr_(2)`
`rArr m+n=1`