Question

The vectors $\mathbf { a } = 2 \lambda ^ { 2 } \mathbf { i } + 4 \lambda \mathbf { j } + \mathbf { k }$ and $\mathbf { b } = 7 \mathbf { i } - 2 \mathbf { j } + \lambda \mathbf { k }$ make an obtuse angle whereas the angle between b and k is acute and less than $\pi / 6$, then domain of $\lambda$ is

• A. $0 < \lambda < \frac { 1 } { 2 }$
• B. $\lambda > \sqrt { 159 }$
• C. $- \frac { 1 } { 2 } < \lambda < 0$
• D. Null set

Answer 1

Answer: (D)

Null set

Answer 2

As angle between $\mathbf { a }$ and $\mathbf { b }$ is obtuse,

̃ $\left( 2 \lambda ^ { 2 } \mathbf { i } + 4 \lambda \mathbf { j } + \mathbf { k } \right) \cdot ( 7 \mathbf { i } - 2 \mathbf { j } + \lambda \mathbf { k } ) < 0$ ̃ $14 \lambda ^ { 2 } - 8 \lambda + \lambda < 0$

̃ $\lambda ( 2 \lambda - 1 ) < 0$ ̃ $0 < \lambda < \frac { 1 } { 2 }$ ......(i)

Angle between $\mathbf { b }$ and $\mathbf { k }$ is acute and less than $\frac { \pi } { 6 }$.

̃ $\lambda = \sqrt { 53 + \lambda ^ { 2 } } \cdot 1 \cdot \cos \theta$

̃ $\cos \theta = \frac { \lambda } { \sqrt { 53 + \lambda ^ { 2 } } }$

$\theta < \frac { \pi } { 6 }$ ̃ $\cos \theta > \cos \frac { \pi } { 6 }$ ̃ $\cos \theta > \frac { \sqrt { 3 } } { 2 }$̃$\frac { \lambda } { \sqrt { 53 + \lambda ^ { 2 } } } > \frac { \sqrt { 3 } } { 2 }$

̃ $4 \lambda ^ { 2 } - 3 \left( 53 + \lambda ^ { 2 } \right) > 0$ ̃ $\lambda ^ { 2 } > 159$ ̃ $\lambda < - \sqrt { 159 }$

or $\lambda > \sqrt { 159 }$ ……(ii)

From (i) and (ii), $\lambda = \phi$. $\therefore$ Domain of $\lambda$ is null set