Coincident, Parallel, Intersecting & Skew Lines (DP IB Analysis & Approaches (AA): HL): Revision Note

Coincident, parallel, intersecting & skew lines

How do I tell if two lines are parallel?

• Two lines are parallel if their direction vectors are scalar multiples

• e.g. $\mathbf{r}=\left(\begin{array}{c}2\\ 1\\ -7\end{array}\right)+{\lambda }_1\left(\begin{array}{c}2\\ 0\\ -8\end{array}\right)$and $\mathbf{r}=\left(\begin{array}{c}1\\ -1\\ 5\end{array}\right)+{\lambda }_2\left(\begin{array}{c}-1\\ 0\\ 4\end{array}\right)$ are parallel

  • This is because $\left(\begin{array}{c}2\\ 0\\ -8\end{array}\right)=-2\left(\begin{array}{c}-1\\ 0\\ 4\end{array}\right)$

How do I tell if two lines are coincident?

• Coincident lines are lines that lie directly on top of each other

  • They are the same line

• Two lines are coincident if:

  • Their direction vectors are scalar multiples

  • Any point on one of the lines also lies on the other

• e.g. $\mathbf{r}=\left(\begin{array}{c}1\\ -8\end{array}\right)+s\left(\begin{array}{c}-4\\ 8\end{array}\right)$ and $\mathbf{r}=\left(\begin{array}{c}-3\\ 0\end{array}\right)+t\left(\begin{array}{c}1\\ -2\end{array}\right)$  are coincident

  • $\left(\begin{array}{c}-4\\ 8\end{array}\right)=-4\left(\begin{array}{c}1\\ -2\end{array}\right)$ so the direction vectors are scalar multiples

  • $\left(\begin{array}{c}1\\ -8\end{array}\right)$is a position vector of a point on the first line

  • $\left(\begin{array}{c}1\\ -8\end{array}\right)=\left(\begin{array}{c}-3\\ 0\end{array}\right)+4\left(\begin{array}{c}1\\ -2\end{array}\right)$ so it also lies on the second line

What are skew lines?

• Two lines are skew if:

  • They are not parallel

  • They do not intersect

• Skew lines are only possible in 3-dimensions

How do I determine whether two non-parallel lines in 3 dimensions are skew or intersecting?

• STEP 1
  Set the vector equations of the two lines equal to each other with different variables

  • e.g. use $\lambda$ and $\mu$ for the parameters

• STEP 2
  Write the three separate equations for the i, j, and k components in terms of $\lambda$ and $\mu$

• STEP 3
  Solve two of the equations to find values for $\lambda$ and $\mu$

• STEP 4
  Check whether the values of $\lambda$ and $\mu$ satisfy the third equation

  • If all three equations are satisfied, then the lines intersect

  • If not all three equations are satisfied, then the lines are skew

How do I find the point of intersection of two intersecting lines?

• Follow the steps above to find the values of $\lambda$ and $\mu$ that satisfy all three equations

• Find the point of intersection by substituting either the value of $\lambda$ or the value of $\mu$ into one of the vector equations