Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{b}=\hat{j}-\hat{k}$. If $\vec{c}$ is a vector such that $\vec{a} \times \vec{c}=\vec{b}$ and $\vec{a} \cdot \vec{c}=3$, then $\vec{a} \cdot(\vec{b} \times \vec{c})$ is equal to

JEE Main · 2021 · 26 Aug 2021 · Shift 1 · Vector Algebra

Let

a = \hat{i} + \hat{j} + \hat{k}

and

b = \hat{j} - \hat{k}

. If

c

is a vector such that

a \times c = b

and

a \cdot c = 3

, then

a \cdot (b \times c)

is equal to

Answer: (A)

The scalar triple product is unchanged by a cyclic shift of its three vectors and changes sign when two of them are swapped, so

a \cdot (b \times c) = b \cdot (c \times a) = - b \cdot (a \times c) = - b \cdot b = - b^{2} .

With

b = \hat{j} - \hat{k}

b^{2} = 2

, so

a \cdot (b \times c) = - 2

JEE Main · 2021 · 26 Aug 2021 · Shift 1 · Single correct option