Question Type 2: Finding the intersection of two planes Exercises

Express the line of intersection of the planes $2x+3y-z=7$ and $x-y+4z=5$ in vector form.

A line is the intersection of the planes $x+3y+z=9$ and $2x-y+4z=1$. Find the projection of the point $(1,0,2)$ onto this line.

Find parametric equations for the line of intersection of the planes $4x+5y+6z=3$ and $x-y+2z=11$.

Write the symmetric equations of the line of intersection of the planes $x+2y+3z=6$ and $2x-y+z=4$.

Find the coordinates where the line of intersection of the planes $x-2y+z=4$ and $3x+y-2z=5$ meets the plane $y=1$.

Calculate the shortest distance from the point $(1,2,3)$ to the line of intersection of the planes $x+y+z=6$ and $2x-y+3z=4$.

Compute the acute angle between the planes $2x-y+z=3$ and $x+y+z=2$, then find the acute angle between their line of intersection and the plane $x+y-z=4$.

Determine the point on the line of intersection of the planes $x+3y-2z=5$ and $2x-y+z=1$ where $y=0$.

Find parametric equations for the line of intersection of the planes $x+2y-z=3$ and $4x-y+5z=7$, then compute the distance from the point $(2,-1,0)$ to this line.

Determine the symmetric form of the line of intersection of the planes $3x-y+2z=8$ and $x+4y-z=1$.

Find the acute angle between the line of intersection of the planes $x-y+z=2$ and $3x+y-z=4$ and the $x$-axis.