Find the angle between the lines $\overrightarrow{r}=\overrightarrow{5i}-\overrightarrow{7j}+\mu (-\overrightarrow{i}+\overrightarrow{4j}+\overrightarrow{2k}) \overrightarrow{r}=-\overrightarrow{2i}+\overrightarrow{k}+\lambda (\overrightarrow{3i}+\overrightarrow{4k})$

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The line $\overrightarrow r=5\overrightarrow i-7\overrightarrow j+ \mu (-\overrightarrow i+4\overrightarrow j+2\overrightarrow k)$ is parallel to $\overrightarrow u = -\overrightarrow i+4\overrightarrow j+2\overrightarrow k$
The line $\overrightarrow r = -2\overrightarrow i+\overrightarrow k+ \lambda (3\overrightarrow i+4\overrightarrow k)$ is parallel to $\overrightarrow v = 3\overrightarrow i+4\overrightarrow k$
The angle between them is $\theta = \cos^{-1} \large\frac{\overrightarrow u.\overrightarrow v}{|\overrightarrow u||\overrightarrow v|}$
$= \cos^{-1} \large\frac{(-1)(3) + 0 +(2)(4)}{\sqrt{1+16+4}\sqrt{9+16}}$
$= \cos^{-1} \large\frac{5}{\sqrt{21}\: 5} = \cos^{-1} \large\frac{1}{\sqrt{21}}$

edited Jun 24, 2013