Cross product with unit vectors The matrix‐like method of performing the cross product with unit vectors—where one aligns the multiplicands in a 3x3 or 3x5 matrix of components—is unnecessarily unwieldy and cumbersome, in my opinion. I use a different method, one I believe I just developed on my own. To illustrate this, let’s use a typical example out of 2‐D kinematics: /
where and cos
/
̂
/
sin
̂ . So cos
/
̂
sin
̂ The order of the cross product matters, so With the cross product, one can factor out any scalars that multiply the entire product. So in the example above cos
/
̂
sin
̂ Now we can perform the cross product. In distributing the pre‐multiplication by , we will have cos
̂ and sin
̂ The first rule to apply is simply that any cross product of two different unit vectors will yield the third unit vector. So ̂
̂
and ̂
̂ and ̂
̂
This is because a property of the cross product is that the result is always perpendicular to the two multiplicands in the cross product. (Worth mentioning also is that if the two multiplicands are not different, then the cross product is 0.) The second rule to apply has to do with setting the sign of the result. There are positive and negative permutations of the cross product. The positive permutations are when the unit vectors are in alphabetical order— ̂, ̂, —with the stipulation that this goes around the corner, so the order , ̂ is a positive permutation too. The cross products shown above are the positive permutations of the unit vectors because the letters are all in alphabetical order with this around‐the‐corner stipulation. The diagram below may also help. If the product is against this positive direction, the result is negative. So ̂
̂
and ̂
̂ and ̂
̂
̂ and sin
̂ Thus for the above example cos
/
cos
sin
̂
̂
cos
̂ ̂ With a little practice this comes naturally, so that one does not have to draw the little permutation diagram or think too long about the cross product of any two unit vectors when performing it. In 2‐D kinematics one also learns to multiply the ̂ product first, because it will yield ̂ as a result, which will make the product in order.