Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that the angle between them is $\frac{\pi}{3}$. If $\lambda \vec{a}+2 \vec{b}$ and $3 \vec{a}-\lambda \vec{b}$ are perpendicular to each other, then the number of values of $\lambda$ in $[-1,3]$ is :

JEE Main · 2025 · 22 Jan 2025 · Shift 2 · Vector Algebra

Let

a

and

b

be two unit vectors such that the angle between them is

\frac{π}{3}

. If

λ a + 2 b

and

3 a - λ b

are perpendicular to each other, then the number of values of

λ

[- 1, 3]

is :

Answer: (D)

Both vectors are unit vectors with

a \cdot b = cos \frac{π}{3} = \frac{1}{2}

.

Perpendicularity gives

(λ a + 2 b) \cdot (3 a - λ b) = 0

3 λ (a \cdot a) - λ^{2} (a \cdot b) + 6 (a \cdot b) - 2 λ (b \cdot b) = 0

3 λ - \frac{λ ^{2}}{2} + 3 - 2 λ = 0 \Rightarrow λ^{2} - 2 λ - 6 = 0

λ = \frac{2 \pm 4 + 24}{2} = 1 \pm 7 .

Since

7 \approx 2.646

, the roots are approximately

3.646

and

- 1.646

, both lying outside

[- 1, 3]

.

The number of values of

λ

[- 1, 3]

is 0.

JEE Main · 2025 · 22 Jan 2025 · Shift 2 · Single correct option