How Do You Find the Angle Between Two Planes or Between a Line and a Plane?

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For a plane written as $ax+by+cz=k$, a normal vector is ((a,b,c)). For a line written as $\mathbf{r}=\mathbf{a}+\lambda\mathbf{d}$, use $\mathbf{d}$ as its direction vector. These vectors may be multiplied by any non-zero scalar without changing the calculated angle, so simplify them before taking the dot product and reduce arithmetic errors.

For two planes, each normal vector is perpendicular to its plane. Therefore, the acute angle between the normal vectors equals the acute angle between the planes. The absolute value ensures that the result is the required acute angle rather than its supplementary angle.

For a line and a plane, let $\phi$ be the angle between the direction vector and the normal. Since the normal is perpendicular to the plane, the required angle is $\alpha=\dfrac{\pi}{2}-\phi$. Consequently, $\sin\alpha=\cos\phi$, producing the line-plane formula.

For example, let two planes have normals $\mathbf{n}_1=(1,1,0)$ and $\mathbf{n}_2=(1,0,1)$. Then

so $\theta=\dfrac{\pi}{3}$.

If a line has direction $\mathbf{d}=(1,1,0)$ and a plane has normal $\mathbf{n}=(1,0,1)$, then

so $\alpha=\dfrac{\pi}{6}$.

A common misconception is to use cosine directly for the line-plane angle. This gives the angle between the line and the normal, not the plane.

Exam technique: Write the relevant vectors before substituting and show the scalar-product formula. On Paper 1, retain exact values where possible; on Paper 2, show the setup before using your GDC. Give non-exact answers to three significant figures, with radians assumed unless degrees are specified.