Find the acute angle between the lines whose direction ratios are… - Vidyadip

Question

Find the acute angle between the lines whose direction ratios are 5, 12, -13 and 3, -4, 5

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Answer

$\cos \theta=\left|\frac{a_1 a_2+b_1 b_2+c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}}\right|$
$=\left|\frac{15-48-65}{\sqrt{25+144+169} \sqrt{9+16+25}}\right|$
$=\left|-\frac{98}{13 \sqrt{2} \times 5 \sqrt{2}}\right|$
$=\left|-\frac{98}{13 \times 5 \times 2}\right|$
$=\frac{49}{65}$
$\theta=\cos ^{-1}\left(\frac{49}{65}\right)$

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