Question Type 1: Finding the point of intersection for a line and a plane Exercises

Determine the point of intersection of the line $(x,y,z)=(2,3,4)+t(-1,0,2)$ with the plane $4x-y+z=10$.

Determine the point where the line $(x,y,z)=(-3,2,1)+t(7,-5,4)$ intersects the plane $3x-y+2z=8$.

Find the intersection of the line $(x,y,z)=(2,0,-1)+t(0,5,2)$ with the plane $2x-3y+z=4$.

Find the coordinates of the point where the line $(x,y,z)=(5,-2,1)+t(3,1,-4)$ meets the plane $6x+2y-3z=7$.

Compute the intersection of the line $(x,y,z)=(1,0,3)+t(-2,2,1)$ with the plane $5x+y-2z=9$.

Find the point of intersection of the line $(x,y,z)=(0,1,-2)+t(3,-2,4)$ with the plane $x-4y+2z=5$.

Find the point of intersection of the line given by $\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 3 \\ 2 \\ 1 \end{pmatrix} + t \begin{pmatrix} 1 \\ 4 \\ 1 \end{pmatrix}$ with the plane $2x+6y+z=1$.

Find the coordinates of the point of intersection of the line $\mathbf{r} = \begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix} + t \begin{pmatrix} 2 \\ 1 \\ 3 \end{pmatrix}$ with the plane $x+2y-3z=4$.

Find the coordinates of the intersection of the line with equation $\mathbf{r} = t \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}$ and the plane with equation $3x - y + 2z = 5$.

Where does the line $\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} + t\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}$ intersect the plane $x+y+z=6$?

Determine where the line $(x,y,z)=(-2,3,5)+t(4,-1,-2)$ meets the plane $-x+2y+3z=1$.

Compute the intersection of the line $(x,y,z)=(-1,4,3)+t(2,-3,1)$ with the plane $x-y+4z=2$.