Question Type 1: Finding the point of intersection for two lines Exercises

Find the coordinates of the point of intersection of the lines $L_1$ and $L_2$ defined by the equations: $L_1: \mathbf{r} = \begin{pmatrix} -2 \\ 1 \\ 3 \end{pmatrix} + t\begin{pmatrix} 4 \\ -3 \\ 2 \end{pmatrix}$ $L_2: \mathbf{r} = \begin{pmatrix} 4 \\ -7 \\ 8 \end{pmatrix} + u\begin{pmatrix} -2 \\ 5 \\ -3 \end{pmatrix}$

Find the point of intersection of the lines $L_1$ and $L_2$ defined by:

$L_1: \mathbf{r} = \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix} + t \begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}$

$L_2: \mathbf{r} = \begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix} + u \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}$

Determine the coordinates of the intersection of the lines $L_1$ and $L_2$ defined by:

$L_1: \mathbf{r} = \begin{pmatrix} 2 \\ 3 \\ 0 \end{pmatrix} + t \begin{pmatrix} 3 \\ -2 \\ 1 \end{pmatrix}$

$L_2: \mathbf{r} = \begin{pmatrix} 12 \\ -7 \\ 1 \end{pmatrix} + u \begin{pmatrix} -1 \\ 4 \\ 2 \end{pmatrix}$

Find the coordinates of the point of intersection of the lines $L_1$ and $L_2$ defined by the following equations:

$L_1: \mathbf{r} = \begin{pmatrix} 0 \\ 1 \\ 3 \end{pmatrix} + t \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}$

$L_2: \mathbf{r} = \begin{pmatrix} 4 \\ 4 \\ 2 \end{pmatrix} + u \begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}$

Determine the coordinates of the intersection of the lines $L_1$ and $L_2$: $L_1: \mathbf{r} = (1, 1, 1) + t(5, 2, -1)$ $L_2: \mathbf{r} = (2, 17, -7) + u(3, -4, 2)$

Determine the coordinates of the point of intersection of the following two lines:

$L_1: \mathbf{r} = \begin{pmatrix} 2 \\ 1 \\ 0 \end{pmatrix} + t\begin{pmatrix} 3 \\ -1 \\ 2 \end{pmatrix}$

$L_2: \mathbf{r} = \begin{pmatrix} 5 \\ -2 \\ 4.5 \end{pmatrix} + u\begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix}$

Find the coordinates of the intersection point.

Find the coordinates of the point of intersection of the two lines.

Find the intersection point of the lines: $L_1: \mathbf{r} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} + t\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}, \quad L_2: \mathbf{r} = \begin{pmatrix} 3 \\ 4 \\ 5 \end{pmatrix} + u\begin{pmatrix} -2 \\ -3 \\ -4 \end{pmatrix}$

Find the coordinates of the point of intersection of the lines $L_1$ and $L_2$.

$L_1: \mathbf{r} = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} + t \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$

$L_2: \mathbf{r} = \begin{pmatrix} 2.5 \\ 3.5 \\ 6 \end{pmatrix} + u \begin{pmatrix} 2 \\ 1 \\ 1 \end{pmatrix}$

Calculate the intersection point of $L_1: (-1,0,4)+t(2,1,3), \quad L_2: (-1,-1,9)+u(1,1,-1).$