Examples

Span of two vectors in 3D

Span of two vectors in ${\mathbb{R}}^3$

Problem: Suppose $\mathbf{u}=(1,2,-1)$ and $\mathbf{v}=(4,-1,3)$. Describe $⟨\mathbf{u},\mathbf{v}⟩$ in vector form.
Solution: By the definition, we have:

$$⟨\mathbf{u},\mathbf{v}⟩=\{(a+4b,2a-b,-a+3b)\mid a,b\in \mathbb{R}\}.$$

To write this in vector form, we can use the cross product to find a vector perpendicular to both $\mathbf{u}$ and $\mathbf{v}$:

$$\begin{array}{rl}\mathbf{n}& =\mathbf{u}\times \mathbf{v}\\ & =(2\cdot 3-(-1)(-1),-(1\cdot 3-4(-1)),1(-1)-4\cdot 2)\\ & =(5,-7,-9).\end{array}$$

Then in vector form the equation is $(x,y,z)\cdot (5,-7,-9)=0$, or in scalar form it is $5x-7y-9z=0$.