A vector V from $L_2$ to $L_1$ and N, the normal to P, are

$\mathbf{V}\=\left[\begin{array}{c}2t-1\+s\\ t-5-s\\ -2-2 t-s\end{array}\right]$  $\mathbf{N}\=\left[\begin{array}{c}1\\ -1\\ 1\end{array}\right]$ 

Loading Student:-MultivariateCalculus

For $k\=1\,2$, define lines $L_k$ as the position vectors ${\mathbf{L}}_k$
Define N, the normal for plane $P$ 

$⟨2 t\+1\,t-2\,2-2 t⟩$$\stackrel{\text{assign to a name}}{\to }$$L_1$

$⟨2-s\,3\+s\,4\+s⟩$$\stackrel{\text{assign to a name}}{\to }$$L_2$

$⟨1\,-1\,1⟩$$\stackrel{\text{assign to a name}}{\to }$$N$

Obtain V, a vector from $L_2$ to $L_1$  

$\mathbf{V}\={\mathbf{L}}_1-{\mathbf{L}}_2$$\stackrel{\text{assign}}{\to }$

Impose the conditions $\mathbf{V}\mathbf{·}\mathbf{N}\=0$ and ${∥\mathbf{V}∥}^2\=54$ and solve the resulting equations

$\mathbf{V}\mathbf{·}\mathbf{N}\=0\,{∥\mathbf{V}∥}^2\=54$

$-t\+2\+s\=0\,{\left(2t-1\+s\right)}^2\+{\left(t-5-s\right)}^2\+{\left(-2-2t-s\right)}^2\=54$

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

$\left\{s\=-3\,t\=-1\right\}\,\left\{s\=0\,t\=2\right\}$

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

$S$

In terms of the parameter $p$, obtain line $L$ as a position vector

$\genfrac{}{}{0ex}{}{\left({\mathbf{L}}_1\+p \mathbf{V}\right)}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{S_1}$ = $$

$\genfrac{}{}{0ex}{}{\left({\mathbf{L}}_1\+p \mathbf{V}\right)}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{S_2}$ = $$

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

For $k\=1\,2$, define lines $L_k$ as the position vectors ${\mathbf{L}}_k$

Define N, the normal for plane $P$ and define $\mathbf{V}\={\mathbf{L}}_1-{\mathbf{L}}_2$ 

${\mathbf{L}}_1\,{\mathbf{L}}_2\,\mathbf{N}\mathbf{\,}\mathbf{V}≔⟨2 t\+1\,t-2\,2-2 t⟩\,⟨2-s\,3\+s\,4\+s⟩\,⟨1\,-1\,1⟩\,{\mathbf{L}}_1-{\mathbf{L}}_2\:$

Via the Norm, DotProduct and solve commands:

Impose and solve the equations arising from the conditions $\mathbf{V}\mathbf{·}\mathbf{N}\=0$ and ${∥\mathbf{V}∥}^2\=54$ 

$S≔\mathrm{solve}\left(\left\{\mathrm{DotProduct}\left(\mathbf{V}\,\mathbf{N}\right)\=0\,\mathrm{Norm}{\left(\mathbf{V}\,\mathrm{Euclidean}\right)}^2\=54\right\}\,\left\{s\,t\right\}\right)$

$\left\{s\=-3\,t\=-1\right\}\,\left\{s\=0\,t\=2\right\}$

Use the eval command to obtain line L

Use the eval command to obtain line L

$⟨x\,y\,z⟩\=\mathrm{eval}\left({\mathbf{L}}_1\+p \mathbf{V}\,S\left[1\right]\right)$

$⟨x\,y\,z⟩\=\mathrm{eval}\left({\mathbf{L}}_1\+p \mathbf{V}\,S\left[2\right]\right)$