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

Step-by-step solution

1. Find direction vector of the line: $\mathbf{d}=(1,2,-1)$.
2. Find normal vector of the plane: $\mathbf{n}=(3,1,-2)$.
3. Compute the angle $\varphi$ between $\mathbf{d}$ and $\mathbf{n}$ using $cosphi=\frac{|\mathbf{d}\cdot \mathbf{n}|}{||\mathbf{d}||,||\mathbf{n}||}$.
4. Dot product: $\mathbf{d}\cdot \mathbf{n}=1\cdot 3+2\cdot 1+(-1)(-2)=7$.
5. Magnitudes: $||\mathbf{d}||=\sqrt{1^2+2^2+(-1{)}^2}=\sqrt{6}$, $||\mathbf{n}||=\sqrt{3^2+1^2+(-2{)}^2}=\sqrt{14}$.
6. Compute cosine: $cosphi=\frac{7}{\sqrt{6}\sqrt{14}}\approx 0.763$.
7. Find $phi={\cos }^{-1}(0.763)\approx 40°$.
8. Angle between line and plane is $90°-phi\approx 50°$.
9. Round to nearest integer: $50°$.

Concept tested: Angle between line and plane

Published 17 September 2026 · How this solution was prepared and checked