Let the cross product of $\mathbf{A}\=a_1 \mathbf{i}\+a_2 \mathbf{j}\+a_3 \mathbf{k}$ and $\mathbf{B}\mathbf{\=}{b}_1 \mathbf{i}\+b_2 \mathbf{j}\+b_3 \mathbf{k}$ be given by the vector $\mathbf{C}\mathbf{\=}{c}_1 \mathbf{i}\+c_2 \mathbf{j}\+c_3 \mathbf{k}$. Solve the three equations

$\mathbf{A}·\mathbf{C}\=a_1{c}_1\+a_2{c}_2\+a_3{c}_3\=0$ 

$\mathbf{B}·\mathbf{C}\=b_1{c}_1\+b_2{c}_2\+b_3{c}_3\=0$ 

$\sqrt{c_1^2\+c_2^2\+c_3^2}\=\sqrt{a_1^2\+a_2^2\+a_3^2}\sqrt{b_1^2\+b_2^2\+b_3^2}\sqrt{1-\frac{{\left(a_1{b}_1\+a_2{b}_2\+a_3{b}_3\right)}^2}{\left(a_1^2\+a_2^2\+a_3^2\right)\left(b_1^2\+b_2^2\+b_3^2\right)}}$

for$$

 $\left[\begin{array}{c}{c}_1\\ c_2\\ c_3\end{array}\right]\=\left[\begin{array}{c}{a}_2{b}_3-a_3{b}_2\\ a_3{b}_1-a_1{b}_3\\ a_1{b}_2-a_2{b}_1\end{array}\right]$ and $\left[\begin{array}{c}{c}_1\\ c_2\\ c_3\end{array}\right]\=\left[\begin{array}{c}-a_2{b}_3\+a_3{b}_2\\ a_1{b}_3-a_3{b}_1\\ -a_1{b}_2\+a_2{b}_1\end{array}\right]$.

The first two equations express the orthogonality of A and B with C, and the third equation expresses the length condition $∥\mathbf{C}∥$ = $∥\mathbf{A}∥$ $∥\mathbf{B}∥$ $\sin \left(\mathrm{\θ}\right)$, where $\mathrm{\θ}\={\cos }^{-1}\left(\frac{\mathbf{A}·\mathbf{B}}{∥\mathbf{A}∥ ∥\mathbf{B}∥}\right)$ is the angle between A and B.

Only one of the two solutions, namely the first, satisfies the right-hand rule. Hence, that is the expression in Definition 1.4.1 for the cross product.

The simplest way to decide which of the two solutions obeys the right-hand rule is to note that the unit basis vectors $\left\{\mathbf{i}\,\mathbf{j}\,\mathbf{k}\right\}$ form a right-handed system. Hence, $\mathbf{i}\times \mathbf{j}\=\mathbf{k}$, and it is just the first of the two solutions shown above that satisfies this relation. (The second solution would yield $\mathbf{i}\times \mathbf{j}\= -\mathbf{k}$.)

Only one of these two solutions obeys the right-hand rule; the one that does is then the cross product.$$

Only one of these two solutions obeys the right-hand rule; the one that does is then the cross product.$$