$\nabla g\left(\mathrm{P}\right)·\mathbf{u}\/∥\mathbf{u}∥\=8$

$\Rightarrow \left[\begin{array}{c}a\\ b\end{array}\right]·\left(\frac{1}{\sqrt{13}}\left[\begin{array}{c}2\\ -3\end{array}\right]\right)\=8$

$\Rightarrow \frac{2 a}{\sqrt{13}}-\frac{3 b}{\sqrt{13}}\=8$

$\nabla g\left(\mathrm{P}\right)·\mathbf{v}\/∥\mathbf{v}∥\=-6$

$\Rightarrow \left[\begin{array}{c}a\\ b\end{array}\right]·\left(\frac{1}{\sqrt{41}}\left[\begin{array}{c}5\\ 4\end{array}\right]\right)\=-6$

$\Rightarrow \frac{5 a}{\sqrt{41}}\+\frac{4 b}{\sqrt{41}}\=-6$

$\nabla g\left(\mathrm{P}\right)·\mathbf{w}\/∥\mathbf{w}∥\=\mathrm{\λ}$

$\Rightarrow \left[\begin{array}{c}a\\ b\end{array}\right]·\(\frac{1}{\sqrt{53}}\left[\begin{array}{c}7\\ -2\end{array}\right]\)\=\mathrm{\λ}$

$\Rightarrow \frac{7 a}{\sqrt{53}}-\frac{2 b}{\sqrt{53}}\=\mathrm{\λ}$

Initialize

Tools≻Load Package: Student Multivariate Calculus

Loading Student:-MultivariateCalculus

$\mathbf{u}\=⟨2\,-3⟩$$\stackrel{\text{assign}}{\to }$

$\mathbf{v}\=⟨5\,4⟩$$\stackrel{\text{assign}}{\to }$

$\mathbf{w}\=⟨7\,-2⟩$$\stackrel{\text{assign}}{\to }$

Obtain and solve the equations $a\=g_x\left(\mathrm{P}\right)\,b\=g_y\left(\mathrm{P}\right)$, and $\mathrm{\λ}\={\mathrm{D}}_{\mathbf{w}}g\left(\mathrm{P}\right)$ 

$\left[\begin{array}{c}a\\ b\end{array}\right]·\frac{\mathbf{u}}{∥\mathbf{u}∥}\=8\,\left[\begin{array}{c}a\\ b\end{array}\right]·\frac{\mathbf{v}}{\∥\mathbf{v}\∥}\=-6\,\left[\begin{array}{c}a\\ b\end{array}\right]·\frac{\mathbf{w}}{∥\mathbf{w}∥}\=\mathrm{\λ}$

$\frac{2}{13}a\sqrt{13}-\frac{3}{13}b\sqrt{13}\=8\,\frac{5}{41}a\sqrt{41}\+\frac{4}{41}b\sqrt{41}\=-6\,\frac{7}{53}a\sqrt{53}-\frac{2}{53}b\sqrt{53}\=\mathrm{\λ}$

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

$\left\{a\=\frac{32}{23}\sqrt{13}-\frac{18}{23}\sqrt{41}\,b\=-\frac{40}{23}\sqrt{13}-\frac{12}{23}\sqrt{41}\,\mathrm{\λ}\=\frac{304}{1219}\sqrt{53}\sqrt{13}-\frac{102}{1219}\sqrt{53}\sqrt{41}\right\}$

$\stackrel{\text{select entry 3}}{\to }$

$\mathrm{\λ}\=\frac{304}{1219}\sqrt{53}\sqrt{13}-\frac{102}{1219}\sqrt{53}\sqrt{41}$

$\stackrel{\text{combine}}{\=}$

$\mathrm{\λ}\=\frac{304}{1219}\sqrt{689}-\frac{102}{1219}\sqrt{2173}$

Initialize

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

$\mathbf{Gg}\,\mathbf{u}\,\mathbf{v}\,\mathbf{w}≔⟨a\,b⟩\,⟨2\,-3⟩\,⟨5\,4⟩\,⟨7\,-2⟩\:$

Form the three equations $\nabla g\left(\mathrm{P}\right)·\mathbf{u}\/∥\mathbf{u}∥\=8\,\nabla g\left(\mathrm{P}\right)·\mathbf{v}\/∥\mathbf{v}∥\=-6\,\nabla g\left(\mathrm{P}\right)·\mathbf{w}\/∥\mathbf{w}∥\=\mathrm{\λ}$ 

$$\begin{gathered} q_1≔\mathrm{DotProduct}\left(\mathbf{Gg}\,\mathrm{Normalize}\left(\mathbf{u}\right)\right)\=8\: \\ {q}_2≔\mathrm{DotProduct}\left(\mathbf{Gg}\,\mathrm{Normalize}\left(\mathbf{v}\right)\right)\=-6\: \\ {q}_3≔\mathrm{DotProduct}\left(\mathbf{Gg}\,\mathrm{Normalize}\left(\mathbf{w}\right)\right)\=\mathrm{\λ}\: \end{gathered}$$

Solve for $a\=g_x\left(\mathrm{P}\right)\,b\=g_y\left(\mathrm{P}\right)$, and $\mathrm{\λ}\={\mathrm{D}}_{\mathbf{w}}g\left(\mathrm{P}\right)$ 

$\mathrm{combine}\left(\mathrm{solve}\left(\left\{q_1\,q_2\,q_3\right\}\,\left\{a\,b\,\mathrm{\λ}\right\}\right)\right)$

$\left\{a\=\frac{32}{23}\sqrt{13}-\frac{18}{23}\sqrt{41}\,b\=-\frac{40}{23}\sqrt{13}-\frac{12}{23}\sqrt{41}\,\mathrm{\λ}\=\frac{304}{1219}\sqrt{689}-\frac{102}{1219}\sqrt{2173}\right\}$