Question

Let $\overrightarrow{{\mathbf{w}}_{\mathbf{1}}}$ and$\overrightarrow{{\mathbf{w}}_{\mathbf{2}}}$betwo nonparallel vectorsin.${\mathit{ℝ}}^{\mathbf{2}}$Considerthe curve $\mathit{C}$ in ${\mathit{ℝ}}^{\mathbf{2}}$thatconsists of all vectors ofthe form$\mathit{c}\mathit{o}\mathit{s}\mathbf{\left(}\mathit{t}\mathbf{\right)}\overrightarrow{{\mathbf{w}}_{\mathbf{1}}}\mathbf{+}\mathit{s}\mathit{i}\mathit{n}\mathbf{\left(}\mathit{t}\mathbf{\right)}\overrightarrow{{\mathbf{w}}_{\mathbf{2}}}$,where$\mathit{t}$isa parameter.Show that$\mathit{C}$ is an ellipse. Hint: You can interpret $\mathit{C}$asthe image of the unit circleunder a suitable linear transformation;then use Exercise $\mathbf{54}$ .

Short answer

It is shown that the image of the unit circle is an ellipse centered at the origin.

Step-by-step solution

Compute the functions

Consider the vector.

$\overrightarrow{v}=\cos \left(t\right)\overrightarrow{v_1}+\sin \left(t\right)\overrightarrow{v_2}$

The linear transformation is given as,

$T\left(\overrightarrow{v}\right)=\cos \left(t\right)T\left(\overrightarrow{v_1}\right)+\sin \left(t\right)T\left(\overrightarrow{v_2}\right)$

Consider the axes 

The axes of the ellipse are,

$\overrightarrow{w_1}=T\left(\overrightarrow{v_1}\right)$

$\overrightarrow{w_2}=T\left(\overrightarrow{v_2}\right)$

The linear transformation is given as,

$T\left(\overrightarrow{w}\right)=\cos \left(t\right)T\left(\overrightarrow{w_1}\right)+\sin \left(t\right)T\left(\overrightarrow{w_2}\right)$

.

Sketch the graph 

The graph of the unit circle and its transformation is,

Hence, the image of unit circle is an ellipse centered at origin.