If vec u , vec v and vecw are three non-coplanar vectors, then ( vec u + vec v- vec w). [( vec u - vec v)× ( vec v - vec w)] equals to

Question

If vec u , vec v and vecw are three non-coplanar vectors, then ( vec u + vec v- vec w). [( vec u - vec v)× ( vec v - vec w)] equals to (A)  u. w× u  (B)  u. v× w  (C)  0  (D)  3 u. v× w

Tardigrade Admin · Accepted Answer

(u+v-w).[(u-v)× (v-w)] =(u+v-w).(u× v-u× w-v× v+v× w) (u+v-w).(u× v-u× w+v× w) [∵ barv× barv=0] =u.(u× v)-u.(u× w)+u.(v× w)+v.(u× v) -v.(u× w)+v.(v× w)-w.(u× v)+w.(u× w) -w.(v× w) =0-0+[u v w]+0-[v u w]+0-[w u v] +0-0 =u.(v× w)

Q. If $u, v an d w$ are three non-coplanar vectors, then $(u + v - w) . [(u - v) \times (v - w)]$ equals to

$u . w \times u$

$u . v \times w$

$0$

$3 u . v \times w$

Q. If u,vandw are three non-coplanar vectors, then (u+v−w).[(u−v)×(v−w)] equals to

u.w×u

u.v×w

0

3u.v×w

(u+v−w).[(u−v)×(v−w)] =(u+v−w).(u×v−u×w−v×v+v×w) (u+v−w).(u×v−u×w+v×w) [∵vˉ×vˉ=0] =u.(u×v)−u.(u×w)+u.(v×w)+v.(u×v) −v.(u×w)+v.(v×w)−w.(u×v)+w.(u×w) −w.(v×w) =0−0+[uvw]+0−[vuw]+0−[wuv] +0−0 =u.(v×w)

Q. If $u, v an d w$ are three non-coplanar vectors, then $(u + v - w) . [(u - v) \times (v - w)]$ equals to

$u . w \times u$

$u . v \times w$

$0$

$3 u . v \times w$