Questions

01

Orthogonality by hand

Determine which pairs of $\mathbf{u}=(2,-1,1)$ and $\mathbf{v}=(2,6,1)$ and $\mathbf{w}=(-4,1,2)$ are orthogonal.

02

Dot product via unit vectors

Compute the dot product of $4{\mathbf{e}}_1-3{\mathbf{e}}_2$ and ${\mathbf{e}}_1+2{\mathbf{e}}_2+{\mathbf{e}}_3$.

03

Vector plane from scalar plane

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

04

Parametric plane from vector plane

Find a parametric equation of the plane passing through ${\mathbf{r}}_0=(3,1,0)$ having normal vector $\mathbf{n}=(3,2,-5)$. (Hint: write the equation for a vector from ${\mathbf{r}}_0$ that lies in the plane, and solve it to find two example vectors.)

05

Is it a span?

Consider the plane $\{(x,y,z)\mid z=1+2x-y\}$. Is this plane the span of two vectors?