$\mathbf{N}\= \nabla f\left(\mathrm{P}\right)\=\left[\begin{array}{c}2\\ 4\\ 6\end{array}\right]$ 

$\left(\mathbf{R}\mathbf{-}\mathbf{P}\right)·\mathbf{N}\=0$

$0$

$\=\left(\left[\begin{array}{c}x\\ y\\ z\end{array}\right]-\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]\right)·\left[\begin{array}{c}2\\ 4\\ 6\end{array}\right]$

$\=2\left(x-1\right)\+4\left(y-1\right)\+6\left(z-1\right)$

$\=2 x\+4 y\+6 z-2-4-6$

$\=2 x\+4 y\+6 z-12$

Initialize

Loading Student:-MultivariateCalculus

Obtain a surface normal at point P

$x^2\+2 y^2\+3 z^2$$\stackrel{\text{gradient}}{\to }$ $\stackrel{\text{select entry 1}}{\to }$ $\stackrel{\text{assign to a name}}{\to }$$N$

Obtain an equation for the tangent plane

$\left[1\&comma;1\&comma;1\right]\&comma;\mathbf{N}$$\stackrel{\text{make plane}}{\to }$$\mathrm{<<\; Plane\; 1\; >>}$$\stackrel{\text{representation}}{\to }$$x\&plus;2y\&plus;3z\&equals;6$

Represent point P as the position vector A

$\mathbf{A}\&equals;⟨1\&comma;1\&comma;1⟩$$\stackrel{\text{assign}}{\to }$

Define the generic position vector R and implement the vector equation of a plane

$\mathbf{R}\&equals;⟨x\&comma;y\&comma;z⟩$$\stackrel{\text{assign}}{\to }$

$\left(\mathbf{R}-\mathbf{A}\right)·\mathbf{N}\&equals;0$

$2x-12\&plus;4y\&plus;6z\&equals;0$

Initialize

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

$f≔x^2\&plus;2 y^2\&plus;3 z^2\&colon;$

Obtain a vector normal to the surface

$\mathbf{N}≔\mathrm{Gradient}\left(f\&comma;\left[x\&comma;y\&comma;z\right]\&equals;\left[1\&comma;1\&comma;1\right]\right)\left[\right]$ = $$

Obtain an equation for the tangent plane

$\mathrm{GetRepresentation}\left(\mathrm{Plane}\left(\left[1\&comma;1\&comma;1\right]\&comma;\mathbf{N}\right)\right)$

$x+2y+3z=6$

$\mathrm{Student}:-\mathrm{VectorCalculus}:-\mathrm{TangentPlane}\left(f\&equals;6\&comma;x\&equals;1\&comma;y\&equals;1\&comma;z\&equals;1\right)$

$2x+4y+6z-12=0$