Let $\overrightarrow{\mathrm{a}}$ be a vector which is perpendicular to the vector $3 \hat{i}+\frac{1}{2} \hat{j}+2 \hat{k}$. If $\vec{a} \times(2 \hat{i}+\hat{k})=2 \hat{i}-13 \hat{j}-4 \hat{k}$, then the projection of the vector $\overrightarrow{\mathrm{a}}$ on the vector $2 \hat{i}+2 \hat{j}+\hat{k}$ is

JEE Main · 2022 · 28 Jun 2022 · Shift 2 · Vector Algebra

Let

a

be a vector which is perpendicular to the vector

3 \hat{i} + \frac{1}{2} \hat{j} + 2 \hat{k}

. If

a \times (2 \hat{i} + \hat{k}) = 2 \hat{i} - 13 \hat{j} - 4 \hat{k}

, then the projection of the vector

a

on the vector

2 \hat{i} + 2 \hat{j} + \hat{k}

Answer: (C)

Use the vector triple product identity:

(a \times (2 \hat{i} + \hat{k})) \times (3 \hat{i} + \frac{1}{2} \hat{j} + 2 \hat{k}) = (2 \hat{i} - 13 \hat{j} - 4 \hat{k}) \times (3 \hat{i} + \frac{1}{2} \hat{j} + 2 \hat{k})

. Since

a ⊥ (3 \hat{i} + \frac{1}{2} \hat{j} + 2 \hat{k})

, we have

- (6 + 2) a = \hat{i} 23 \hat{j} - 13 \frac{1}{2} \hat{k} - 4 2

. Solving gives

a = 3 \hat{i} + 2 \hat{j} - 5 \hat{k}

. The projection on

2 \hat{i} + 2 \hat{j} + \hat{k}

\frac{a \cdot ( 2 i ^ + 2 j ^ + k ^ )}{∣2 i ^ + 2 j ^ + k ^ ∣} = \frac{6 + 4 - 5}{3} = \frac{5}{3}

JEE Main · 2022 · 28 Jun 2022 · Shift 2 · Single correct option