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

Question

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

Tardigrade Admin · Accepted Answer

( vecu+ vecv- vecw)⋅(( vecu- vecv)×( vecv- vecw)) =( vecu+ vecv- vecw)⋅( vecu× vecv- vecu× vecw+ vecv× vecw) [∵ vecv× vecv= vec0] = vecu⋅ vecv× vecw- vecv⋅ vecu× vecw- vecw⋅ vecu× vecv [All other variables] =[ vecu vecv vecw]+[ vecu vecv vecw]-[ vecu vecv vecw] =[ vecu vecv vecw]

Q. If $u, v$ and $w$ are three non coplanar vectors, then $(u + v - w) . [(u - v) \times (v - w)] =$

0

$u . v \times w$

$u . w \times v$

$3 u . v \times w$

$(u + v - w) \cdot ((u - v) \times (v - w))$
$= (u + v - w) \cdot (u \times v - u \times w + v \times w)$
$[∵ v \times v = 0]$
$= u \cdot v \times w - v \cdot u \times w - w \cdot u \times v$
[All other variables]
$= [u v w] + [u v w] - [u v w]$
$= [u v w]$

Q. If u,v and w are three non coplanar vectors, then (u+v−w).[(u−v)×(v−w)]=

0

u.v×w

u.w×v

3u.v×w

(u+v−w)⋅((u−v)×(v−w)) =(u+v−w)⋅(u×v−u×w+v×w) [∵v×v=0] =u⋅v×w−v⋅u×w−w⋅u×v [All other variables] =[uvw]+[uvw]−[uvw] =[uvw]

Q. If $u, v$ and $w$ are three non coplanar vectors, then $(u + v - w) . [(u - v) \times (v - w)] =$

$u . v \times w$

$u . w \times v$

$3 u . v \times w$

$(u + v - w) \cdot ((u - v) \times (v - w))$
$= (u + v - w) \cdot (u \times v - u \times w + v \times w)$
$[∵ v \times v = 0]$
$= u \cdot v \times w - v \cdot u \times w - w \cdot u \times v$
[All other variables]
$= [u v w] + [u v w] - [u v w]$
$= [u v w]$