$\mathbf{A}·\left(\mathbf{A}\times \mathbf{B}\right)$

$\=$ $\left[\begin{array}{c}{a}_1\\ a_2\\ a_3\end{array}\right]·\left[\begin{array}{c}{a}_2{b}_3-a_3{b}_2\\ a_3{b}_1-a_1{b}_3\\ a_1{b}_2-a_2{b}_1\end{array}\right]$

$\=a_1\left(a_2{b}_3-a_3{b}_2\right)\+a_2\left(-a_1{b}_3\+a_3{b}_1\right)\+a_3\left(a_1{b}_2-a_2{b}_1\right)$

$\=0$

$\mathbf{B}\mathbf{·}\left(\mathbf{A}\times \mathbf{B}\right)$

$\=$ $\left[\begin{array}{c}{b}_1\\ b_2\\ b_3\end{array}\right]·\left[\begin{array}{c}{a}_2{b}_3-a_3{b}_2\\ a_3{b}_1-a_1{b}_3\\ a_1{b}_2-a_2{b}_1\end{array}\right]$$$

$\=b_1\left(a_2{b}_3-a_3{b}_2\right)\+b_2\left(-a_1{b}_3\+a_3{b}_1\right)\+b_3\left(a_1{b}_2-a_2{b}_1\right)$

$\=0$

Initialize

Loading Student:-MultivariateCalculus

$⟨a_1\,a_2\,a_3⟩$$\stackrel{\text{assign to a name}}{\to }$

$⟨b_1\,b_2\,b_3⟩$$\stackrel{\text{assign to a name}}{\to }$

Compute $\mathbf{A}·\left(\mathbf{A}\times \mathbf{B}\right)$ and $\mathbf{B}·\left(\mathbf{A}\times \mathbf{B}\right)$ 

$\mathbf{A}·\left(\mathbf{A}\times \mathbf{B}\right)$ = $a_1\left(a_2{b}_3-a_3{b}_2\right)\+a_2\left(-a_1{b}_3\+a_3{b}_1\right)\+a_3\left(a_1{b}_2-a_2{b}_1\right)$$\stackrel{\text{simplify}}{\=}$$0$$$

$\mathbf{B}·\left(\mathbf{A}\times \mathbf{B}\right)$ = $b_1\left(a_2{b}_3-a_3{b}_2\right)\+b_2\left(-a_1{b}_3\+a_3{b}_1\right)\+b_3\left(a_1{b}_2-a_2{b}_1\right)$$\stackrel{\text{simplify}}{\=}$$0$$$

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

$\mathbf{A}\,\mathbf{B}≔⟨a_1\,a_2\,a_3⟩\,⟨b_1\,b_2\,b_3⟩\:$

$\mathrm{simplify}\left(\mathrm{DotProduct}\left(\mathbf{A}\,\mathrm{CrossProduct}\left(\mathbf{A}\,\mathbf{B}\right)\right)\right)$ = $0$$$