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Solve the following.
Find the angle between the lines and the plane .
Find the angle between the lines $\frac{x+3}{3} = \frac{y-1}{2} = \frac{z-2}{4}$ and the plane $2x + y - 3z + 4 = 0$.
Answer
Answer
AIDirection of the line $\vec{b} = 3\hat{i} + 2\hat{j} + 4\hat{k}$; normal to the plane $\vec{n} = 2\hat{i} + \hat{j} - 3\hat{k}$.
$\sin\theta = \frac{|\vec{b} \cdot \vec{n}|}{|\vec{b}||\vec{n}|} = \frac{|6 + 2 - 12|}{\sqrt{29}\sqrt{14}} = \frac{4}{\sqrt{406}}$
Hence $\theta = \sin^{-1}\left(\frac{4}{\sqrt{406}}\right)$.
Final answer: $\sin^{-1}\left(\frac{4}{\sqrt{406}}\right)$
From ISC 2026 Improvement Mathematics Paper 1, question 15(v).
Check your working with the 3D geometry and vectors calculator: points, vectors, lines and planes: distances, angles, foot and image, shortest distance, intersections.