Exercises

01

Vector plane from scalar plane

Find the vector equation of a plane with scalar equation $7x-4y+2z+10=0$.

02

Scalar plane from vector plane

Find the scalar equation of a plane passing through ${\mathbf{r}}_0=(3,1,0)$ and with normal vector $\mathbf{n}=(3,2,-5)$. (Start with the vector form.)

03

Line and plane intersection

Find the point at which the plane $3x-9y+2z-7=0$ and the line $\mathbf{r}(t)=(1,2,1)+t(-2,0,1)$ intersect.

04

Angle between planes

Find a formula for the angle formed between two planes. Use this formula to find the angle between the planes passing through $(0,0,0)$ with normal vectors ${\mathbf{n}}_1=(1,2,1)$ and ${\mathbf{n}}_2=(4,1,3)$, respectively.

05

Distance between parallel planes

Find the distance from the plane given by $15x+3y-3z-10=0$ to the parallel plane given by $5x+y-z-3=0$.