Before we dive into the examples, there is one last item. Throughout the coordinate systems and vector properties/components sections, have you wondered how the x, y, and z-coordinates are related? They are related by something known as the right hand rule. With your right hand, point your fingers flat out. If you were to point your fingers towards the along the x-axis, and then curl your fingers towards the y-axis, your thumb will point in the direction of the z-axis. This is something that you will see all the time if you study electricity and magnetism. Current, electric, and magnetic fields all relate in direction by the right hand rule.
Cross products follow the right hand rule: i-hat cross j-hat is k-hat. Of course, i-hat is the unit vector pointing in the x-direction, j-hat aligns with the y-direction and k-hat the z-direction. Following the axes, j-hat cross k-hat is i-hat. You could use your right hand rule to prove this.
Here is a trick: Think of the letters ijkijk. If you run to the right, any combination of three letters yields a positive cross product, i x j = k, j x k = i, k x i = j. If you run the letters to the left, any combination of three letters yields a negative cross product: k x j = -i, j x i = -k, and so on. This can give you a general idea of which direction your cross product vector is pointing. Now, on with the examples.
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