The line $L$ is given by the parametric equations $x=1+2t,y=-1+t,z=3-t$. Find the acute angle between this line and the plane $4x+y-2z+5=0$. Give your answer in degrees rounded to the nearest integer.

Step-by-step solution

1. Identify the direction vector of the line: $\mathbf{v}=(2,1,-1)$.
2. Identify the normal vector of the plane: $\mathbf{n}=(4,1,-2)$.
3. Find the angle ${\theta }^{\prime }$ between $\mathbf{v}$ and $\mathbf{n}$ using $\cos {\theta }^{\prime }=\frac{\mathbf{v}\cdot \mathbf{n}}{|\mathbf{v}||\mathbf{n}|}$.
4. Compute the dot product: $\mathbf{v}\cdot \mathbf{n}=2\cdot 4+1\cdot 1+(-1)(-2)=8+1+2=11$.
5. Compute magnitudes: $|\mathbf{v}|=\sqrt{2^2+1^2+(-1{)}^2}=\sqrt{6}$, $|\mathbf{n}|=\sqrt{4^2+1^2+(-2{)}^2}=\sqrt{21}$.
6. Hence $\cos {\theta }^{\prime }=\frac{11}{\sqrt{6}\sqrt{21}}=\frac{11}{\sqrt{126}}\approx 0.980$.
7. Find ${\theta }^{\prime }={\cos }^{-1}(0.980)\approx 11.5^{\circ }$.
8. The angle $\theta$ between the line and the plane is $90^{\circ }-{\theta }^{\prime }\approx 78.5^{\circ }$.
9. Round to the nearest integer: $\theta =79^{\circ }$.

Concept tested: Angle between line and plane

Published 4 September 2026 · How this solution was prepared and checked