Let a, b and c be three non-zero vectors such that no two of them are collinear and (a × b) × c = 1/3|b||c|a. If θ is the angle between vectors b and c, then a value of sinθ is

Let a, b and c be three non-zero vectors such that no two… — JEE Main Mathematics PYQ

JEE Main

Let

\vec{a}

\vec{b}

and

\vec{c}

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

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

.

If

\theta

is the angle between vectors

\vec{b}

and

\vec{c}

, then a value of

\sin\theta

A.
$\frac{2\sqrt{2}}{3}$
B.
$\frac{-\sqrt{2}}{3}$
C.
$\frac{2}{3}$
D.
$\frac{-2\sqrt{3}}{3}$

Show answer and solution

Correct option: A

Solution

Using the BAC-CAB rule: (a × b) × c = (a·c)b - (b·c)a.

Setting this equal to (1/3)|b||c|a:

(a·c)b - (b·c)a = (1/3)|b||c|a

Since a and b are non-collinear, comparing coefficients:
- Coefficient of b: a·c = 0 (so a ⊥ c)
- Coefficient of a: -(b·c) = (1/3)|b||c|

From the second equation: b·c = -(1/3)|b||c|, which means |b||c|cosθ = -(1/3)|b||c|, so cosθ = -1/3.

Therefore sinθ = √(1 - 1/9) = √(8/9) = 2√2/3.

(Taking positive value since θ is between 0 and π, sinθ ≥ 0.)

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