The acute angle between the line joining the points (2,1,-3), (-3… - Vidyadip

MCQ

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

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

Correct option: A.

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

a
(a) Direction ratio of the line joining the point $(2,\,\,1,\,\, - 3),\,$ $\,( - \,3,\,\,1,\,\,7)$ are $({a_1},\,\,{b_1},\,\,{c_1})\,$

$\, \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 $({a_2},\,{b_2},\,\,{c_2})\,\, $

$\Rightarrow \,\,(3,\,\,4,\,\,5)$

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} }}\,\, $

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

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