Problems

01

Line of intersection between planes

Find a parametric equation for the line formed as the intersection of the two planes $6x-3y+z-5=0$ and $-x+y+5z-5=0$. What is the angle between these planes?

02

Span

Find $a,b,c$ for the scalar equation $ax+by+cz=0$ of a plane which is equal to $⟨\mathbf{u},\mathbf{v}⟩$, where $\mathbf{u}=(2,0,1)$ and $\mathbf{v}=(-1,1,3)$.

03

Independence

Is this a dependent set of vectors?:

$$\left\{\left(\begin{array}{c}1\\ -2\\ 3\end{array}\right),\left(\begin{array}{c}4\\ 0\\ 2\end{array}\right),\left(\begin{array}{c}3\\ -2\\ 4\end{array}\right)\right\}$$

04

Distance point to line

Find the distance from the point $\mathbf{p}=(1,5,-2)$ to the line given by $\mathbf{r}(t)=(-2,3,1)+t(4,0,-1)$.