The Vector or Cross Product

In Fig. 1 are shown two vectors $𝐚$ and $𝐛$, which make an angle measured in the plane of the two vectors. The vector product $𝐚\times 𝐛$ is defined as a vector along line $OB$ normal to the plane of the vectors $𝐚$ and $𝐛$, with such a direction that, if one looks in the direction $OB$, a clockwise turn will bring the direction of $𝐚$ into that of $𝐛$. The rule for direction may also be stated by saying that the direction is that of the advance of a right-handed screw turned from $𝐚$ to $𝐛$. The magnitude of the vector product is defined as $(a)(b)\sin \theta$. The magnitude is thus equal to twice the area of the triangle shown shaded in Fig. 1. Because of the form in which it is written the vector product $𝐚\times 𝐛$ is often called the cross product.

From the above definition it is seen that the vector product depends upon the order in which the vectors are taken. The vector product $𝐛\times 𝐚$ would have the same magnitude as $𝐚\times 𝐛$, but would have the opposite direction so that $$𝐛\times 𝐚=-𝐚\times 𝐛$$ i.e., vector products are not commutative.

Writing the vector product in the form of components, we have: \begin{aligned} \mathbf{a} \times \mathbf{b} =& (a_x \mathbf{i} + a_y \mathbf{j} + a_z \mathbf{k}) \times (b_x \mathbf{i} + b_y \mathbf{j} + b_z \mathbf{k})\\ =& a_x b_z (\mathbf{i} \times \mathbf{i}) + a_x b_y (\mathbf{i} \times \mathbf{j}) + a_x b_z (\mathbf{i} \times \mathbf{k}) \\ &+ a_y b_x (\mathbf{j} \times \mathbf{i}) + a_y b_y (\mathbf{j} \times \mathbf{j}) + a_y b_z (\mathbf{j} \times \mathbf{k}) \\ &+ a_z b_x (\mathbf{k} \times \mathbf{i}) + a_z b_y (\mathbf{k} \times \mathbf{j}) + a_z b_z (\mathbf{k} \times \mathbf{k}) \end{aligned}

From the definition of the vector product, we have the following relations between the unit vectors $𝐢,𝐣,𝐤$: $$𝐢\times 𝐢=𝐣\times 𝐣=𝐤\times 𝐤=0$$ $$𝐢\times 𝐣=-𝐣\times 𝐢=𝐤$$ $$𝐢\times 𝐤=-𝐤\times 𝐢=-𝐣$$ $$𝐣\times 𝐤=-𝐤\times 𝐣=𝐢$$ hence: $$𝐚\times 𝐛=(a_y{b}_z-a_z{b}_y)𝐢+(a_z{b}_x-a_x{b}_z)𝐣+(a_x{b}_y-a_y{b}_x)𝐤$$