Solveeit Logo
Login

Question

Question: The angle between the lines \(r = ( 4 i - j ) + s ( 2 i + j - 3 k )\) and \(r = ( i - j + 2 k ) +...

The angle between the lines r=(4i−j)+s(2i+j−3k)r = ( 4 i - j ) + s ( 2 i + j - 3 k ) and r=(i−j+2k)+t(i−3j+2k)r = ( i - j + 2 k ) + t ( i - 3 j + 2 k ) is

A

3π2\frac { 3 \pi } { 2 }

B

π3\frac { \pi } { 3 }

C

2π3\frac { 2 \pi } { 3 }

D

π6\frac { \pi } { 6 }

Answer

π3\frac { \pi } { 3 }

Explanation

Solution

We have, r=(4i−j)+s(2i+j−3k)\mathbf { r } = ( 4 \mathbf { i } - \mathbf { j } ) + s ( 2 \mathbf { i } + \mathbf { j } - 3 \mathbf { k } ) and

r=(i−j+2k)+t(i−3j+2k)\mathbf { r } = ( \mathbf { i } - \mathbf { j } + 2 \mathbf { k } ) + t ( \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k } )

We know that, ,

, cos⁡θ=−12\cos \theta = - \frac { 1 } { 2 }

Hence, acute angle θ=cos⁡−1(12)\theta = \cos ^ { - 1 } \left( \frac { 1 } { 2 } \right) i.e. θ=π3\theta = \frac { \pi } { 3 }