Let $\hat{u}$ and $\hat{v}$ be unit vectors inclined at an acute angle such that $|\hat{\mathrm{u}} \times \hat{\mathrm{v}}|=\frac{\sqrt{3}}{2}$. If $$\overrightarrow{\mathrm{A}}=\lambda \hat{\mathrm{u}}+\hat{\mathrm{v}}+(\hat{\mathrm{u}} \times \hat{\mathrm{v}})$$ Then $\lambda$ is equal to :

(A) $\frac{4}{3}(\vec{A} \cdot \hat{u})-\frac{2}{3}(\vec{A} \cdot \hat{v})$

(B) $\frac{2}{3}(\vec{A} \cdot \hat{u})-\frac{1}{3}(\vec{A} \cdot \hat{v})$

(C) $\frac{4}{3}(\vec{A} \cdot \hat{u})+\frac{2}{3}(\vec{A} \cdot \hat{v})$

(D) $(\vec{A} \cdot \hat{u})-\frac{1}{2}(\vec{A} \cdot \hat{v})$

Vector Algebra — JEE Main · 2026 · 04 Apr 2026 · Shift 2 Question with Solution

Let

\overset{u}{^}

and

\overset{v}{^}

be unit vectors inclined at an acute angle such that

∣ \overset{u}{^} \times \overset{v}{^} ∣ = \frac{3}{2}

. If

A = λ \overset{u}{^} + \overset{v}{^} + (\overset{u}{^} \times \overset{v}{^})

Then

λ

is equal to :

(A) $\frac{4}{3} (A \cdot \overset{u}{^}) - \frac{2}{3} (A \cdot \overset{v}{^})$ ✓ Correct
(B) $\frac{2}{3} (A \cdot \overset{u}{^}) - \frac{1}{3} (A \cdot \overset{v}{^})$
(C) $\frac{4}{3} (A \cdot \overset{u}{^}) + \frac{2}{3} (A \cdot \overset{v}{^})$
(D) $(A \cdot \overset{u}{^}) - \frac{1}{2} (A \cdot \overset{v}{^})$

Answer: (A)

Since

∣ \overset{u}{^} \times \overset{v}{^} ∣ = sin θ = \frac{3}{2}

and the angle is acute,

θ = \frac{π}{3}

and

\overset{u}{^} \cdot \overset{v}{^} = cos \frac{π}{3} = \frac{1}{2}

. Dot

A = λ \overset{u}{^} + \overset{v}{^} + (\overset{u}{^} \times \overset{v}{^})

with

\overset{u}{^}

: since

\overset{u}{^} \cdot \overset{u}{^} = 1

\overset{u}{^} \cdot \overset{v}{^} = \frac{1}{2}

and

\overset{u}{^} \cdot (\overset{u}{^} \times \overset{v}{^}) = 0

, this gives

A \cdot \overset{u}{^} = λ + \frac{1}{2}

. Dot with

\overset{v}{^}

A \cdot \overset{v}{^} = \frac{λ}{2} + 1

. From these,

2 (A \cdot \overset{u}{^}) = 2 λ + 1

and

A \cdot \overset{v}{^} = \frac{λ}{2} + 1

; forming

2 (A \cdot \overset{u}{^}) - (A \cdot \overset{v}{^}) = 2 λ + 1 - \frac{λ}{2} - 1 = \frac{3 λ}{2}

. Hence

λ = \frac{2}{3} (2 (A \cdot \overset{u}{^}) - (A \cdot \overset{v}{^})) = \frac{4}{3} (A \cdot \overset{u}{^}) - \frac{2}{3} (A \cdot \overset{v}{^})

JEE Main · 2026 · 04 Apr 2026 · Shift 2 · Single correct option