Equation of a Plane in Vector Form (DP IB Analysis & Approaches (AA): HL): Revision Note

Equation of a plane in vector form

How do I find the vector equation of a plane?

• The formula for finding the vector equation of a plane is $\mathit{r}=\mathit{a}+\lambda \mathit{b}+\mu \mathit{c}$

  • $\mathit{r}$ is the position vector of any point on the plane

  • $\mathit{a}$ is the position vector of a known point on the plane

  • $\mathit{b}$ and $\mathit{c}$are two non-parallel direction vectors which are parallel to the plane

  • $\lambda$ and $\mu$ are scalars

• A plane in often denoted using the capital Greek letter $\Pi$

• There are an infinite number of ways to write the equation

  • There are an infinite number of options for $\mathit{a}$

  • There are an infinite number of pairs non-parallel direction vectors

  • Any scalar multiple of a direction vector is also a direction vector

How do I find the vector equation of a plane that passes through three points?

• Suppose a line passes through the points with position vectors $\mathit{a}$, $\mathit{p}$ and $\mathit{q}$

• Find two direction vectors

  • For example, $\mathit{a}-\mathit{p}$, $\mathit{a}-\mathit{q}$ or $\mathit{p}-\mathit{q}$

    • It is important that all three points do not lie on the same line

• Use the given formula

  • For example, $\mathit{r}=\mathit{a}+\lambda \left(\mathit{a}-\mathit{p}\right)+\mu \left(\mathit{a}-\mathit{q}\right)$

How do I determine whether a point lies on a plane?

• Write the components of the vectors in the equation

  • $\mathit{r}\mathit{ }=\mathit{ }\left(\genfrac{}{}{0pt}{}{a_1}{\begin{array}{c}{a}_2\\ a_3\end{array}}\right)+\lambda \left(\genfrac{}{}{0pt}{}{b_1}{\begin{array}{c}{b}_2\\ b_3\end{array}}\right)+\mu \left(\genfrac{}{}{0pt}{}{c_1}{\begin{array}{c}{c}_2\\ c_3\end{array}}\right)$

• Write the components of the position vector of the point to test

  • $\mathit{p}=\mathit{ }\left(\genfrac{}{}{0pt}{}{p_1}{\begin{array}{c}{p}_2\\ p_3\end{array}}\right)$ 

• Form a system of linear equations

  • $p_1= a_1+\lambda b_1+\mu c_1$

  • $p_2= a_2+\lambda b_2+\mu c_2$

  • $p_3= a_3+\lambda b_3+\mu c_3$

• Solve two of the equations to find a value of 

• Check that these values also satisfy the third equation

  • If they do, then the point lies on the line

  • Otherwise, the point does not lie on the line