Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three vectors such that $\vec{a}=\vec{b} \times(\vec{b} \times \vec{c})$. If magnitudes of the vectors $\vec{a}, \vec{b}$ and $\vec{c}$ are $\sqrt{2}, 1$ and 2 respectively and the angle between $\vec{b}$ and $\vec{c}$ is $\theta\left(0<\theta<\frac{\pi}{2}\right)$, then the value of $1+\tan \theta$ is equal to :

(B) $\frac{\sqrt{3}+1}{\sqrt{3}}$

JEE Main · 2021 · 27 Jul 2021 · Shift 2 · Vector Algebra

Let

a, b

and

c

be three vectors such that

a = b \times (b \times c)

. If magnitudes of the vectors

a, b

and

c

are

2, 1

and 2 respectively and the angle between

b

and

c

θ (0 < θ < \frac{π}{2})

, then the value of

1 + tan θ

is equal to :

Answer: (C)

The vector

b

is perpendicular to

b \times c

, so taking magnitudes in

a = b \times (b \times c)

∣ a ∣ = ∣ b ∣ b \times c = ∣ b ∣^{2} ∣ c ∣ sin θ

Substituting the given magnitudes,

2 = 1^{2} \cdot 2 sin θ

, so

sin θ = \frac{1}{2}

. With

0 < θ < \frac{π}{2}

this gives

θ = \frac{π}{4}

and

tan θ = 1

, so

1 + tan θ = 2

JEE Main · 2021 · 27 Jul 2021 · Shift 2 · Single correct option