Find the angle between the pair of lines r = (3 i + j - 2 k) + ( … - Vidyadip

Question

Find the angle between the pair of lines
$\vec r = (3\hat i + \hat j - 2\hat k) + \lambda (\hat i - \hat j -2\hat k)$ and $\vec r = (2\hat i - \hat j - 56\hat k) + \mu (3\hat i - 5\hat j - 4\hat k)$

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Answer

Let $\theta $ be the angle between the given lines
${\vec b_1} = \hat i - \hat j + 2\hat k$ and ${\vec b_2} = 3\hat i - 5\hat j - 4\hat k$
$\cos \theta = \left| {\frac{{{{\vec b}_1}.{{\vec b}_2}}}{{\left| {{{\vec b}_1}} \right|\left| {{{\vec b}_2}} \right|}}} \right|$
$\left| {\frac{{\left( {\hat i - \hat j - 2\hat k} \right).\left( {3\hat i - 5\hat j - 4\hat k} \right)}}{{\left| {\hat i - \hat j - 2\hat k} \right|\left| {3\hat i - 5\hat j - 4\hat k} \right|}}} \right|$
$=|\frac{3+5+8}{\sqrt{1^2+1^2+2^2}\sqrt{3^2+5^2+4^2}}|=\frac{16}{\sqrt6\sqrt{50}}=\frac{16}{\sqrt{2\times3}\sqrt{2\times25}}$
$= \frac{16}{10\times\sqrt3}$
$=\frac{8}{5\sqrt3}$
$\Rightarrow\theta=\cos^{-1}\left(\frac{8}{5\sqrt3}\right)$

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