Question

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

• A. $\frac{3}{4}$
• B. $\frac{1}{2}$
• C. $-\frac{1}{4}$
• D. $\frac{1}{4}$

Answer 1

Answer: (D)

$\frac{1}{4}$

Answer 2

(a⋅c)b−(a⋅b)c=2b−c​
a⋅c=21​,a⋅b=21​
∴b⋅d=21​
(a×b)⋅(c×d)=a⋅(b×(c×d))
=a⋅((b⋅d)c−(b⋅c)d)
=(a⋅c)(b⋅d)=41​