Question

If three unit vectors a, b, c are such that $\mathbf { a } \times ( \mathbf { b } \times \mathbf { c } ) = \frac { \mathbf { b } } { 2 }$ then the vector a makes with b and c respectively the angles

• A. $40 ^ { \circ } , 80 ^ { \circ }$
• B. $45 ^ { \circ } , 45 ^ { \circ }$
• C. $30 ^ { \circ } , 60 ^ { \circ }$
• D. $90 ^ { \circ } , 60 ^ { \circ }$

Answer 1

Answer: (D)

$90 ^ { \circ } , 60 ^ { \circ }$

Answer 2

As we know, ......(i)

$\because \mathbf { a } \times ( \mathbf { b } \times \mathbf { c } ) = \frac { \mathbf { b } } { 2 }$ (Given)

From equation (i),

or $\left( \mathbf { a } . \mathbf { c } - \frac { 1 } { 2 } \right) \mathbf { b } - ( \mathbf { a } \cdot \mathbf { b } ) \mathbf { c } = \mathbf { c }$

Comparison on both sides of b and

$\mathbf { a } \cdot \mathbf { c } - \frac { 1 } { 2 } = 0$

$\Rightarrow | \mathbf { a } \| \mathbf { c } | \cos \theta = \frac { 1 } { 2 } \Rightarrow ( 1 ) ( 1 ) \cos \theta = \frac { 1 } { 2 } \Rightarrow \theta = 60 ^ { \circ }$

or , $\therefore \theta = 90 ^ { \circ }$.

So the angle between $\mathbf { a }$ with b and c are $90 ^ { \circ }$ and $60 ^ { \circ }$ respectively.