Let veca, vecb and vecc be three non zero vectors such that vecb ⋅ vecc=0 and veca ×( vecb × vecc)=( vecb- vecc/2). If vecd be a vector such that vecb ⋅ vecd= veca ⋅ vecb, then ( veca × vecb) ⋅( vecc × vecd) is equal to

Question

Let veca, vecb and vecc be three non zero vectors such that vecb ⋅ vecc=0 and veca ×( vecb × vecc)=( vecb- vecc/2). If vecd be a vector such that vecb ⋅ vecd= veca ⋅ vecb, then ( veca × vecb) ⋅( vecc × vecd) is equal to (A) (3/4) (B) (1/2) (C) -(1/4) (D) (1/4)

Tardigrade Admin · Accepted Answer

( veca ⋅ vecc) vecb-( veca ⋅ vecb) vec c =( vecb- vecc/2) veca ⋅ vecc=(1/2), veca ⋅ vecb=(1/2) ∴ vecb ⋅ vecd=(1/2) ( veca × vecb) ⋅( vecc × vecd)= veca ⋅( vecb ×( vecc × vecd)) = veca ⋅(( vecb ⋅ vecd) vecc-( vecb ⋅ vecc) vecd) =( veca ⋅ vecc)( vecb ⋅ vecd)=(1/4)

Q. Let $a, b$ and $c$ be three non zero vectors such that $b \cdot c = 0$ and $a \times (b \times c) = \frac{b - c}{2}$ . If $d$ be a vector such that $b \cdot d = a \cdot b$ , then $(a \times b) \cdot (c \times d)$ is equal to

$\frac{3}{4}$

$\frac{1}{2}$

$- \frac{1}{4}$

$\frac{1}{4}$

$(a \cdot c) b - (a \cdot b) c = \frac{b - c}{2}$
$a \cdot c = \frac{1}{2}, a \cdot b = \frac{1}{2}$
$∴ b \cdot d = \frac{1}{2}$
$(a \times b) \cdot (c \times d) = a \cdot (b \times (c \times d))$
$= a \cdot ((b \cdot d) c - (b \cdot c) d)$
$= (a \cdot c) (b \cdot d) = \frac{1}{4}$

Q. Let a,b and c be three non zero vectors such that b⋅c=0 and a×(b×c)=2b−c​. If d be a vector such that b⋅d=a⋅b, then (a×b)⋅(c×d) is equal to

43​

21​

−41​

41​