# Let      be three non-zero vectors such that no two of them are collinear and   if  is the angle between vectors then a value of   is: Option 1) Option 2) Option 3) Option 4)

V Vakul

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)

This option is correct.

Option 2)

This option is incorrect.

Option 3)

This option is incorrect.

Option 4)

This option is incorrect.

Exams
Articles
Questions