Angles Between a Line & a Plane (DP IB Analysis & Approaches (AA): HL): Revision Note

Angle between line & plane

What is meant by the angle between a line and a plane?

• The angle between a line and a plane is defined to be the angle between:

  • The line

  • The projection of the line onto the plane

    • This is the line of intersection between the plane and a perpendicular plane which contains the line

• This is the smallest angle between the line and the plane

• It is easiest to think of these two lines making a right-triangle with the normal vector to the plane

  • The line joining the plane will be the hypotenuse

  • The line on the plane will be adjacent to the angle

  • The normal will the opposite the angle

How do I find the angle between a line and a plane?

• For example, consider:

  • The line with equation $l:\mathit{r}=\left(\begin{array}{c}3\\ -1\\ -2\end{array}\right)+\lambda \left(\begin{array}{c}1\\ -2\\ 0\end{array}\right)$

  • The plane with equation $\Pi :2x+3y-z=5$

• STEP 1

  Find the acute angle between the direction vector of the line and the normal vector to the plane

  • Use the formula $\cos \alpha =\frac{|\mathit{b}·\mathit{n}|}{\left|\mathit{b}\right|\left|\mathit{n}\right|}$

    • $\cos \alpha =\frac{\left|\left(\begin{array}{c}1\\ -2\\ 0\end{array}\right)·\left(\begin{array}{c}2\\ 3\\ -1\end{array}\right)\right|}{\left|\left(\begin{array}{c}1\\ -2\\ 0\end{array}\right)\right|\left|\left(\begin{array}{c}2\\ 3\\ -1\end{array}\right)\right|}=\frac{4}{\sqrt{70}}$

    • $\alpha ={\cos }^{-1}\left(\frac{4}{\sqrt{70}}\right)=61.439...°$

• STEP 2
  Subtract this angle from 90° to find the acute angle between the line and the plane

  • Subtract the angle from $\frac{\mathrm{\pi }}{2}$ if working in radians

  • $\theta =90°-61.439...°=28.560...°$