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Addition of Vectors

Let a , b a...

Question

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

A

\frac{2}{3}

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B

\frac{- 2 \sqrt{3}}{3}

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C

\frac{2 \sqrt{2}}{3}

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D

\frac{- \sqrt{2}}{3}

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Solution

The correct option is B $\frac{2 \sqrt{2}}{3}$
$\begin{matrix} (\to a \times \to b) \times \to c = - c \times (\to c \times \to b) = - [(\to c . \to b) \to a - (\to c . \to a) \to b] = (\to c . \to a) \to b - (\to c . \to b) \to a \equiv \frac{1}{3} ∣ ∣ \to b ∣ ∣ | \to c | | \to a | - (\to c . \to b) = \frac{1}{3} ∣ ∣ \to b ∣ ∣ | \to c | - | \to c | ∣ ∣ \to b ∣ ∣ cos θ = \frac{1}{3} ∣ ∣ \to b ∣ ∣ | \to c | cos θ = \frac{1}{3} sin θ = \frac{2 \sqrt{2}}{3} H e n c e, o p t i o n C i s c o r r e c t a n s w e r . \end{matrix}$

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