Angles using vectors (HL)

One formula, but with lots of asterisks.

The core idea is to use the $\cos\theta$ in dot products.

Contents

• Angle between lines
  • exception: vertices
• Angle between two planes
  • exception: faces and half-planes
• Angle between line and planes

Angle between lines

The angle between lines is same as the angle between their direction vectors $\bm {d_1}$ and $\bm {d_2}$, so from dot product:

exception: vertices

If finding $A\hat BC$, use $\overrightarrow{BA}$ and $\overrightarrow{BC}$ as the direction vectors, to ensure you get the correct angle $\theta$, and not $\pi - \theta$. This is because for lines, both $\theta$ and $\pi-\theta$ should be accepted. But for a named angle specifically, only this is correct.

Angle between two planes

Similar to angles between two lines, the formula is

for the two normal vectors $\bm {n_1}$ and $\bm {n_2}$ of the two planes.

exception: faces and half-planes

For a polyhedron, if the normals both point outwards or both inwards, then the angle between faces is $\bm{\pi - \theta}$, this is what happens in ~95% of such questions.

In very rare circumstances when one normal points outwards and other points inwards, then the angle between the faces is $\theta$.

See this math stack exchange answer. 

Angle between line and planes

First find the angle between the normal $\bm n$ and the direction vector of the line $\bm d$

However, the normal is perpendicular to the plane, so we need to take the complementary angle, by subtracting from $\displaystyle \frac\pi2$.

The actual answer is