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Let $\vec{a},\vec{b},\; and\; \vec{c}$ be non-zero vectors such that $(\vec{a}\times \vec{b})\times \vec{c}=\frac{1}{3}\left |\,\vec{b} \, \right |\left |\vec{c} \, \right |\vec{a}.$ If $\Theta$ is the acute angle between the vectors $\vec{b}\;\: and\;\; \vec{c},then\; \sin \Theta$ equals

Option 1)
$\frac{2}{3}\;$

Option 2)
$\; \frac{\sqrt{2}}{3}\;$

Option 3)
$\; \frac{1}{3}\;$

Option 4)
$\; \frac{2\sqrt{2}}{3}$

Answers (1)

best_answer

As we have learned

Vector Triple Product (VTP) -

$\vec{a}\times \left ( \vec{b} \times \vec{c}\right )= \left ( \vec{a}.\vec{c} \right )\vec{b}-\left ( \vec{a}.\vec{b}\right )\vec{c}$

$\left ( \vec{a}\times \vec{b} \right )\times \vec{c}= \left ( \vec{a}.\vec{c} \right )\vec{b}-\left ( \vec{b}.\vec{c}\right )\vec{a}$

- wherein

$\vec{a}, \vec{b}, \vec{c}$ are three vectors.

Scalar Product of two vectors (dot product) -

$\vec{a}\vec{b}=\left | a \right |\left | b \right |Cos\theta$

- wherein

$\Theta$ is the angle between the vectors $\vec{a}\: and\:\vec{b}$

We have

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

$(\vec{a}\cdot \vec{c})\vec{b}-(\vec{b}\cdot \vec{c})\vec{a} = 1/3 |\vec{b}||\vec{c}|\vec{a}$

$( \vec{b}\cdot \vec{c}) = -1/3$

$\cos \theta = -1/3$

$\sin \theta = \sqrt{1- 1/9 } = 2 \frac{\sqrt 2}{3}$

Option 1)

$\frac{2}{3}\;$

Option 2)

$\; \frac{\sqrt{2}}{3}\;$

Option 3)

$\; \frac{1}{3}\;$

Option 4)

$\; \frac{2\sqrt{2}}{3}$

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Himanshu

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