What Is the Vector Product?

The Vector product (cross product) is a new way to multiply vectors, where the answer is also a vector! I think the absolute easiest way to calculate the cross product is to cross in towers:

Don’t try to remember the letter combinations, that is really cumbersome and unnecessary. Rather, try to remember the pattern you follow when crossing the vectors. You will discover that $y_1{z}_2-y_2{z}_1$ forms a cross when you multiply like in the formula:

$$\begin{array}{ccc}\hfill (\hfill & \hfill y_1,\hfill & \hfill z_1)\hfill \\ \hfill \hfill & \hfill \times \hfill & \hfill \hfill \\ \hfill (\hfill & \hfill y_2,\hfill & \hfill z_2)\hfill \end{array}$$

The same goes for $z_1{x}_2-z_2{x}_1$:

$$\begin{array}{ccc}\hfill (x_1,\hfill & \hfill \hfill & \hfill z_1)\hfill \\ \hfill \hfill & \hfill \times \hfill & \hfill \hfill \\ \hfill (x_2,\hfill & \hfill \hfill & \hfill z_2)\hfill \end{array}$$

And it also goes for $x_1{y}_2-x_2{y}_1$:

$$\begin{array}{ccc}\hfill (x_1,\hfill & \hfill y_1,\hfill & \hfill )\hfill \\ \hfill \hfill & \hfill \times \hfill & \hfill \hfill \\ \hfill (x_2,\hfill & \hfill y_2,\hfill & \hfill )\hfill \end{array}$$

Notice that the next cross begins with the variable that the last cross ended with. That means you only need to remember to begin in the middle and move towards the right. Make up two vectors and try the method out, you won’t regret it!

If you use this formula, you can find the length of a cross product if you know the angle $\alpha$ between the two vectors:

If you have the cross product as a vector, you can use the formula for vector length.