Intersections of Two Planes (DP IB Analysis & Approaches (AA): HL): Revision Note

Intersection of two planes

How can I tell if two planes are parallel?

• Two planes are parallel if their normal vectors are scalar multiples

  • A normal vector for the plane $\mathit{r}=\mathit{a}+\lambda \mathit{b}+\mu \mathit{c}$ is $\mathit{n}=\mathit{b}\times \mathit{c}$

• You can spot this easier if the equations are written in Cartesian form $ax+by+cz=d$

  • Planes are parallel if the left-hand side of the equations are scalar multiples

    • For example, $2x+3y+z=5$ and $4x+6y+2z=9$ are parallel

    • For example, $2x+3y+z=5$ and $4x+6y+z=9$ are not parallel

How do I find the line of intersection of two non-parallel planes?

• Two non-parallel planes intersect at a line

  • This is different from two lines which intersect at a point

• To find the line of intersection you need to write the equations of the planes in Cartesian form

• For example, consider the planes with equations   

  • ${\Pi }_1:2x-y+3z=7$

  • ${\Pi }_2:x-3y+4z=11$

Using algebra

• STEP 1
  Choose one variable and substitute this variable for $\lambda$ in both equations

  • Using $x=\lambda$

    • $2\mathrm{\lambda }-y+3z=7$

    • $\mathrm{\lambda }-3y+4z=11$

• STEP 2
  Rearrange the two equations to isolate the $\lambda$ terms and the constant terms on one side

  •  $y-3z=2\lambda -7$

  • $3y-4z=\lambda -11$

• STEP 3
  Solve the equations simultaneously to find the two variables in terms of $\lambda$

  • $z=2-\lambda$

  • $y=-1-\lambda$

    • If there are no solutions, then choose a different variable to be $\lambda$

• STEP 4
  Write the three parametric equations for $x$, $y$, and $z$ in terms of $\lambda$

  • $x=\lambda$

  • $y=-1-\lambda$

  • $z=2-\lambda$

Using the normal vector

• STEP 1
  Find a direction vector of the line by taking the vector product of the two normal vectors of the planes

  • $\left(\begin{array}{c}2\\ -1\\ 3\end{array}\right)\times \left(\begin{array}{c}1\\ -3\\ 4\end{array}\right)=\left(\begin{array}{c}5\\ -5\\ -5\end{array}\right)$

• STEP 2
  Find a point that lies on both planes by setting one of $x$, $y$, and $z$ equal to zero and solving the equations simultaneously to find the other two

  • Using $z=0$

    • $2x-y=7$

    • $x-3y=11$

    • $x=2$ and $y=-3$

• STEP 3
  Write the equation in the form $\mathit{r}=\mathit{a}+\lambda \mathit{b}$ where $\mathit{a}$ is the position vector of the point on both planes and $\mathit{b}$ is the direction vector

  • $\mathit{r}=\left(\begin{array}{c}2\\ -3\\ 0\end{array}\right)+\lambda \left(\begin{array}{c}5\\ -5\\ -5\end{array}\right)$