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JEE Main · 2023 · 24 Jan 2023 · Shift 2 · Vector Algebra

Let

u = \hat{i} - \hat{j} - 2 k, v = 2 \hat{i} + \hat{j} - k, v \cdot w = 2

and

v \times w = u + λ v

. Then

u \cdot w

is equal to

(A)1✓ Correct
(B) $\frac{3}{2}$
(C)2
(D) $- \frac{2}{3}$

Answer: (A)

Solution

Dotting

v \times w = u + λ v

with

w

and using

w \cdot (v \times w) = 0

gives

0 = u \cdot w + λ (v \cdot w) = u \cdot w + 2 λ

. Dotting instead with

v

and using

v \cdot (v \times w) = 0

,

u \cdot v = 3

and

∣ v ∣^{2} = 6

gives

0 = 3 + 6 λ

, so

λ = - \frac{1}{2}

. Hence

u \cdot w = - 2 λ = 1.

JEE Main · 2023 · 24 Jan 2023 · Shift 2 · Single correct option

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