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Loading Student:-MultivariateCalculus

Obtain the line through the origin and along V

$\left[0\&comma;0\&comma;0\right]\&comma;⟨6\&comma;1\&comma;-13⟩$$\stackrel{\text{make line}}{\to }$$\mathrm{<<\; Line\; 2\; >>}$$\stackrel{\text{assign to a name}}{\to }$$L$

Obtain the projection of U onto the line along V

$\left[3\&comma;-5\&comma;7\right]\&comma;L$ = $\left[3\&comma;-5\&comma;7\right]\&comma;\mathrm{<<\; Line\; 2\; >>}$$\stackrel{\text{projection}}{\to }$$\left[-\frac{234}{103}\&comma;-\frac{39}{103}\&comma;\frac{507}{103}\right]$$\stackrel{\text{to Vector}}{\to }$ $$

Obtain the position-vector form for line $L$ 

Write the name $L$.

Context Panel: Evaluate and Display Inline

Context Panel: Student Multivariate Calculus≻Lines & Planes≻Representation

Context Panel: Assign to a Name≻R

$L$ = $\mathrm{<<\; Line\; 2\; >>}$$\stackrel{\text{representation}}{\to }$$\left[\begin{array}{c}6t\\ t\\ -13t\end{array}\right]$$\stackrel{\text{assign to a name}}{\to }$$R$

Obtain $d$, the distance from U to line $L$ 

$⟨3\&comma;-5\&comma;7⟩-\mathbf{R}$

$\stackrel{\text{Euclidean-norm}}{\to }$

$\sqrt{{\left|-3\&plus;6t\right|}^2\&plus;{\left|t\&plus;5\right|}^2\&plus;{\left|7\&plus;13t\right|}^2}$

$\stackrel{\text{assuming real}}{\to }$

$\sqrt{206t^2\&plus;156t\&plus;83}$

$\stackrel{\text{assign to a name}}{\to }$

$d$

Minimize $d$

Calculus palette: Differentiation operator
Press the Enter key.

Context Panel: Solve≻Solve

Context Panel: Assign to a Name≻$T$

$\frac{\&DifferentialD;}{\&DifferentialD; t} d\&equals;0$

$\frac{1}{2}\frac{412t\&plus;156}{\sqrt{206t^2\&plus;156t\&plus;83}}\&equals;0$

$\stackrel{\text{solve}}{\to }$

$\left\{t\&equals;-\frac{39}{103}\right\}$

$\stackrel{\text{assign to a name}}{\to }$

$T$

Evaluate R at the minimizing value of $t$ 

$\genfrac{}{}{0ex}{}{\mathbf{R}}{\phantom{x\&equals;a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{T}$ = $$

Initialize

$\mathrm{with}\left(\mathrm{Student}:-\mathrm{MultivariateCalculus}\right)\&colon;$

$\mathbf{U}\&comma;\mathbf{V}≔⟨3\&comma;-5\&comma;7⟩\&comma;⟨6\&comma;1\&comma;-13⟩\&colon;$

Apply the Projection command to obtain the projection of U onto V.

$\mathrm{Projection}\left(\mathbf{U}\&comma;\mathbf{V}\right)$

$d≔\mathrm{simplify}\left(\mathrm{Student}:-\mathrm{LinearAlgebra}:-\mathrm{Norm}\left(U-t V\right)\right)$ = $\sqrt{206t^2\&plus;156t\&plus;83}$$$

$\mathrm{\&tau;}≔\mathrm{solve}\left(\mathrm{diff}\left(d\&comma;t\right)\&equals;0\&comma;t\right)$

$-\frac{39}{103}$

$\mathrm{\&tau;} V$ = $$