Find the acute angle (in degrees, rounded to the nearest integer) between the line given by $x=1+3t$, $y=-2+4t$, $z=2-t$ and the plane $2x-y+3z=5$.

Step-by-step solution

1. Direction vector of the line: $\mathbf{v}=(3,4,-1)$.
2. Normal to the plane: $\mathbf{n}=(2,-1,3)$.
3. Compute $\mathbf{v}\cdot \mathbf{n}=3\cdot 2+4\cdot (-1)+(-1)\cdot 3=-1$.
4. Magnitudes: $|\mathbf{v}|=\sqrt{9+16+1}=\sqrt{26}$, $|\mathbf{n}|=\sqrt{4+1+9}=\sqrt{14}$.
5. Cosine of the angle between $\mathbf{v}$ and $\mathbf{n}$: $\cos \theta =\frac{|\mathbf{v}\cdot \mathbf{n}|}{|\mathbf{v}||\mathbf{n}|}=\frac{1}{\sqrt{26}\sqrt{14}}\approx 0.0524$.
6. Angle $\theta \approx {\cos }^{-1}(0.0524)\approx 87.0°$.
7. Angle between line and plane $=90°-\theta \approx 3.0°$.

Concept tested: angle line–plane

Published 19 September 2026 · How this solution was prepared and checked