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

Angle between two vectors

What is meant by the angle between two vectors?

• The angle between two vectors is the angle formed by drawing both vectors from the same starting point

How do I find the angle between two vectors?

• You can find the angle between two vectors by using the formula $\mathit{v}·\mathit{w}=\left|\mathit{v}\right|\left|\mathit{w}\right|\cos \theta$

• This can be rearranged to give $\cos \theta = \frac{v_1{w}_1+ v_2{w}_2+ v_3{w}_3}{\left|\mathit{v}\right|\left|\mathit{w}\right|}$

  • $\mathit{v}= \left(\genfrac{}{}{0pt}{}{v_1}{\begin{array}{c}{v}_2\\ v_3\end{array}}\right)$

  • $\mathit{w}= \left(\genfrac{}{}{0pt}{}{w_1}{\begin{array}{c}{w}_2\\ w_3\end{array}}\right)$

• To find the angle between two vectors:

  • Calculate the scalar product between them

  • Calculate the magnitude of each vector

  • Use the formula to find $\cos \theta$

  • Use inverse trig to find $\theta$

• 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

Perpendicular vectors

How do I know if two vectors are perpendicular?

• Two vectors are perpendicular if their scalar product is zero

  • This is because the angle between two perpendicular vectors is 90°

  • $\cos 90°=0$ which means $\mathit{v}·\mathit{w}=0$

• The converse is also true

  • If $\mathit{v}·\mathit{w}=0$ for non-zero vectors

  • Then $\mathit{v}$ and $\mathit{w}$ must be perpendicular