Let $\vec{a}=3 \hat{i}+2 \hat{j}+2 \hat{k}$ and $\vec{b}=\hat{i}+2 \hat{j}-2 \hat{k}$ be two vectors. If a vector perpendicular to both the vectors $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$ has the magnitude 12 then one such vector is:

(A) $4(2 \hat{i}-2 \hat{j}-\hat{k})$

(B) $4(2 \hat{i}-2 \hat{j}+\hat{k})$

(C) $4(2 \hat{i}+2 \hat{j}+\hat{k})$

(D) $4(2 \hat{i}+2 \hat{j}-\hat{k})$

JEE Main · 2019 · 12 Apr 2019 · Shift 1 · Vector Algebra

Let

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

and

b = \hat{i} + 2 \hat{j} - 2 \hat{k}

be two vectors. If a vector perpendicular to both the vectors

a + b

and

a - b

has the magnitude 12 then one such vector is:

Answer: (A)

First form

a + b = 4 \hat{i} + 4 \hat{j}

and

a - b = 2 \hat{i} + 4 \hat{k}

. A vector perpendicular to both is their cross product,

(a + b) \times (a - b) = \hat{i} 42 \hat{j} 40 \hat{k} 04 = 16 \hat{i} - 16 \hat{j} - 8 \hat{k} = 8 (2 \hat{i} - 2 \hat{j} - \hat{k})

. Its magnitude is

8 4 + 4 + 1 = 24

, so to make the length

12

we halve it, giving

4 (2 \hat{i} - 2 \hat{j} - \hat{k})

as one such vector.

JEE Main · 2019 · 12 Apr 2019 · Shift 1 · Single correct option