If `vec(p)!= vec(0)` and the conditions `vec(p).vec(q)=vec(p).vec(r) and vec(p)xxvec(q)=vec(p)xxvec(r)` hold simultaneously, then which one of the following is correct?

Options

A. `vec(q) !=vec(r)`
B. `vec(q)=-vec(r)`
C. `vec(q).vec(r)=0`
D. `vec(q)=vec(r)`

Answer

D

Explanation

Given that `vec(p).vec(q)=vec(p).vec(r)`
`rArr vec(p).(vec(q)-vec(r))=0`
`rArr vec(p)` is perpendicular to `vec(q)-vec(r)`
Also , `vec(p)xxvec(q)=vec(p)xxvec(r)` (given).
`rArr vec(p)xx(vec(q)-vec(r))=0`
`rArr vec(p) ` is parallel to `vec(q)-vec(r)`
Which is not possible simultaneously unless either `vec(p) or vec(q)-vec(r) ` is zero, since `vec(p) != 0, rArr vec(q)-vec(r)=0`
Thus, the given conditions hold simultaneously if `vec(q)=vec(r)`.