Angle Between Two Lines (DP IB Analysis & Approaches (AA): HL): Revision Note

Angle between two lines

How do we find the angles between two lines?

• The angles between two lines are equal to the angles between their direction vectors

  • They can be found using the scalar product of their direction vectors

• The intersection of two lines will always create two pairs of equal angles

  • one pair are acute

  • the other are obtuse

• You can find one angle between $\mathit{r}={\mathit{a}}_1+\lambda {\mathit{b}}_1$ and $\mathit{r}={\mathit{a}}_2+\lambda {\mathit{b}}_2$ using $\theta ={\cos }^{-1}\left(\frac{{\mathit{b}}_1\mathit{ }\mathbf{·}{\mathit{ }\mathit{b}}_2}{\left|{\mathit{b}}_1\right|\left|{\mathit{ }\mathit{b}}_2\right|}\right)$

• You can find the other angle by subtracting this from 180°

  • Or from $\pi$ if you are working in radians

• The sign of the scalar product determines whether $\theta$ is acute or obtuse

  • It is acute if the scalar product is positive

  • It is obtuse if the scalar product is negative

• You can always find the acute angle first by taking the absolute value of the scalar product

  •  ${\cos }^{-1}\left(\frac{\left|{\mathit{b}}_{\mathbf{1}} ·{\mathbf{ }\mathbf{b}}_{\mathbf{2}}\right|}{\left|{\mathit{b}}_1\right|\left|{\mathit{ }\mathit{b}}_2\right|}\right)$