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

Equation of a line in vector form

How do I find the vector equation of a line?

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

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

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

  • $\mathit{b}$ is a direction (displacement) vector

  • $\lambda$ is a scalar

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

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

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

How do I find the vector equation of a line that passes through two points?

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

• Find a direction vector

  • Both $\mathit{a}-\mathit{p}$ and $\mathit{p}-\mathit{a}$ are direction vectors

• Use the given formula

  • $\mathit{r}=\mathit{a}\pm \lambda \left(\mathit{a}-\mathit{p}\right)$

  • $\mathit{r}=\mathit{p}\pm \lambda \left(\mathit{a}-\mathit{p}\right)$

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

• To check if the position vector $\mathit{p}$ lies on the line $\mathit{r}=\mathit{a}+\lambda \mathit{b}$

  • Substitute $\mathit{r}=\mathit{p}$ into the equation

    • $\mathit{p}=\mathit{a}+\lambda \mathit{b}$

  • Check to see if there is a value of $\lambda$ which makes the equation true

  • This is the same as checking if $\mathit{p}-\mathit{a}$ is a scalar multiple of $\mathit{b}$

    • $\mathit{p}-\mathit{a}=\lambda \mathit{b}$

• You can also use algebra

  • Write the components of the vectors in the equation

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

  • 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

    • $p_1= a_1+\lambda b_1$

    • $p_2= a_2+\lambda b_2$

    • $p_3= a_3+\lambda b_3$

  • Solve one of the equations to find a value of $\lambda$

  • Check that this value also satisfies the other two equations

    • If it does, then the point lies on the line

    • Otherwise, the point does not lie on the line