Let   \overrightarrow{a},\overrightarrow{b}and\, \overrightarrow{c}   be three non-zero vectors such that no two of them are collinear and

(\overrightarrow{a}\times \overrightarrow{b})\times \overrightarrow{c}=\frac{1}{3}\left | \overrightarrow{b} \right |\, \left | \overrightarrow{c} \right |\,\overrightarrow{a}.   if  \theta is the angle between vectors \overrightarrow{b}\, and \, \overrightarrow{c},

then a value of \sin \theta  is:

  • Option 1)

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

  • Option 2)

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

  • Option 3)

    \frac{2}{3}

  • Option 4)

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

 

Answers (1)

As we learnt in 

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.

 

and

 

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}

 

\left ( \underset{a}{\rightarrow}\times \underset{b}{\rightarrow} \right )\times \underset{c}{\rightarrow}=\frac{1}{3}\left | \underset{b}{\rightarrow} \right |\left | \underset{c}{\rightarrow} \right |\underset{a}{\rightarrow}

\Rightarrow \left ( \underset{a}{\rightarrow}\cdot \underset{c}{\rightarrow} \right )\underset{b}{\rightarrow}-\left ( \underset{b}{\rightarrow}\cdot \underset{c}{\rightarrow} \right )\underset{a}{\rightarrow}=\frac{1}{3}\left | \underset{b}{\rightarrow} \right |\left | \underset{c}{\rightarrow} \right |\left | \underset{a}{\rightarrow} \right |

 \Rightarrow \left ( \underset{a}{\rightarrow}\cdot \underset{c}{\rightarrow} \right )\underset{b}{\rightarrow}=\left ( \frac{1}{3}\left | \underset{b}{\rightarrow} \right |\left | \underset{c}{\rightarrow} \right |+\underset{b}{\rightarrow}\cdot \underset{c}{\rightarrow} \right )\underset{a}{\rightarrow}

\underset{a}{\rightarrow}\cdot \underset{c}{\rightarrow}=0\:\:and\:\:\frac{1}{3}\left | \underset{b}{\rightarrow} \right |\left | \underset{c}{\rightarrow} \right |+\underset{b}{\rightarrow}\cdot \underset{c}{\rightarrow}=0

Thus

\left | \underset{b}{\rightarrow} \right |\left | \underset{c}{\rightarrow} \right |\left ( \frac{1}{3}+cos\theta \right )=0

cos\theta =\frac{-1}{3}

sin\theta =\frac{2\sqrt{2}}{3}

 

 


Option 1)

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

This option is correct.

Option 2)

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

This option is incorrect.

Option 3)

\frac{2}{3}

This option is incorrect.

Option 4)

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

This option is incorrect.

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