Let $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{c}}$ be three non-zero vectors such that no two of them are collinear and $(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}) \times \overrightarrow{\mathrm{c}}=\frac{1}{3}|\overrightarrow{\mathrm{~b}}||\overrightarrow{\mathrm{c}}| \overrightarrow{\mathrm{a}}$. If $\theta$ is the angle between vectors $\vec{b}$ and $\vec{c}$, then a value of $\sin \theta$ is :

JEE Main · 2015 · 04 Apr 2015 · Offline · Vector Algebra

Let

a, b

and

c

be three non-zero vectors such that no two of them are collinear and

(a \times b) \times c = \frac{1}{3} ∣ b ∣∣ c ∣ a

. If

θ

is the angle between vectors

b

and

c

, then a value of

sin θ

is :

Answer: (D)

Open up the triple product with

(a \times b) \times c = (a \cdot c) b - (b \cdot c) a

, which is to equal

\frac{1}{3} ∣ b ∣∣ c ∣ a

. Since

a

and

b

are not parallel, the

b

term must vanish, forcing

a \cdot c = 0

, and matching the

a

terms gives

- (b \cdot c) = \frac{1}{3} ∣ b ∣∣ c ∣

. Hence

cos θ = - \frac{1}{3}

, so

sin θ = 1 - \frac{1}{9} = \frac{2 2}{3}

JEE Main · 2015 · 04 Apr 2015 · Offline · Single correct option