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

JEE Main · 2023 · 25 Jan 2023 · Shift 1 · Vector Algebra

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

Answer: (D)

By the triple-product identity

a \times (b \times c) = (a \cdot c) b - (a \cdot b) c

, comparison with

\frac{b - c}{2}

gives

a \cdot c = \frac{1}{2}

and

a \cdot b = \frac{1}{2}

, so

b \cdot d = a \cdot b = \frac{1}{2}

. Using

(a \times b) \cdot (c \times d) = (a \cdot c) (b \cdot d) - (a \cdot d) (b \cdot c)

together with

b \cdot c = 0

, the value is

\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}

JEE Main · 2023 · 25 Jan 2023 · Shift 1 · Single correct option