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Q.
The angle between the line $\frac{x + 1}{3} = \frac{y - 1}{2} = \frac{z - 2}{4}$ and the plane $2x + y - 3 \, z + 4 = 0$ is
Introduction to Three Dimensional Geometry
Solution:
The d.n, of the line are $< 3, 2, 4 >$ and attitude numbers of the plane are $< 2, 1, - 3 >$.
Hence, the angle between the line and the plane is given by
$\sin\theta = \frac{\left|3 \times2+2 \times1+4 \times\left(-3\right)\right|}{\sqrt{3^{2}+2^{2}+4^{2}}\sqrt{2^{2} +1^{2}+3^{2}}} $
i.e., by $\sin\theta = \frac{4}{\sqrt{29}\sqrt{14}} = \frac{4}{\sqrt{406}}$