Question

The acute angle between the line joining the points (2,1,–3), (–3,1,7) and a line parallel to $\frac { x - 1 } { 3 } =$ $\frac { y } { 4 } = \frac { z + 3 } { 5 }$ through the point (–1, 0, 4) is

• A. $\cos ^ { - 1 } \left( \frac { 7 } { 5 \sqrt { 10 } } \right)$
• B. $\cos ^ { - 1 } \left( \frac { 1 } { \sqrt { 10 } } \right)$
• C. $\cos ^ { - 1 } \left( \frac { 3 } { 5 \sqrt { 10 } } \right)$
• D. $\cos ^ { - 1 } \left( \frac { 1 } { 5 \sqrt { 10 } } \right)$

Answer 1

Answer: (A)

$\cos ^ { - 1 } \left( \frac { 7 } { 5 \sqrt { 10 } } \right)$

Answer 2

Direction ratio of the line joining the point $( 2,1 , - 3 )$ $( - 3,1,7 )$ are $\left( a _ { 1 } , b _ { 1 } , c _ { 1 } \right)$

$\Rightarrow ( - 3 - 2,1 - 1,7 - ( - 3 ) ) \Rightarrow ( - 5,0,10 )$

Direction ratio of the line parallel to line $\frac { x - 1 } { 3 } = \frac { y } { 4 } = \frac { z + 3 } { 5 }$ are $\left( a _ { 2 } , b _ { 2 } , c _ { 2 } \right) \Rightarrow ( 3,4,5 )$

Angle between two lines,

$\cos \theta = \frac { a _ { 1 } a _ { 2 } + b _ { 1 } b _ { 2 } + c _ { 1 } c _ { 2 } } { \sqrt { a _ { 1 } ^ { 2 } + b _ { 1 } ^ { 2 } + c _ { 1 } ^ { 2 } } \sqrt { a _ { 2 } ^ { 2 } + b _ { 2 } ^ { 2 } + c _ { 2 } ^ { 2 } } }$

$\cos \theta = \frac { ( - 5 \times 3 ) + ( 0 \times 4 ) + ( 10 \times 5 ) } { \sqrt { 25 + 0 + 100 } \sqrt { 9 + 16 + 25 } }$

$\cos \theta = \frac { 35 } { 25 \sqrt { 10 } } \Rightarrow \theta = \cos ^ { - 1 } \left( \frac { 7 } { 5 \sqrt { 10 } } \right)$ .