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JEE Main · 2022 · 27 Jul 2022 · Shift 1 · Vector Algebra

Let

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

and

b = α \hat{i} + β \hat{j} + 2 \hat{k}

. If

((a \times b) \times \hat{i}) \cdot \hat{k} = \frac{23}{2}

, then

∣ b \times 2 \hat{j} ∣

is equal to

(A)4
(B)5✓ Correct
(C) $21$
(D) $17$

Answer: (B)

Solution

Using the vector triple product identity:

((a \times b) \times \hat{i}) = (a \cdot \hat{i}) b - (b \cdot \hat{i}) a

.

Taking the dot product with

\hat{k}

:

(a \cdot \hat{i}) (b \cdot \hat{k}) - (b \cdot \hat{i}) (a \cdot \hat{k}) = \frac{23}{2}

Substituting:

2 \times 2 - α \times 5 = \frac{23}{2}

Solving:

4 - 5 α = \frac{23}{2} \Rightarrow α = - \frac{3}{2}

Now,

b \times 2 \hat{j} = - 4 \hat{i} + 2 α \hat{k}

Therefore,

∣ b \times 2 \hat{j} ∣ = 16 + 4 α^{2} = 16 + 9 = 5

JEE Main · 2022 · 27 Jul 2022 · Shift 1 · Single correct option

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