Encyclopedia>Numbers and Quantities>Vectors>Three Dimensions>Vector Calculations>How Does the Right-hand Rule for Cross Products Work?

When you cross two vectors, $\overrightarrow{a}\times \overrightarrow{b}$, you know that you will get a vector that is perpendicular to both $\overrightarrow{a}$ and $\overrightarrow{b}$. If you study the illustration below, you will notice that $\overrightarrow{a}$ and $\overrightarrow{b}$ are pointing in different directions. To know exactly which direction $\overrightarrow{a}\times \overrightarrow{b}$ is pointing, you use the right-hand rule. It goes like this:

Rule

- 1.
- Create a thumbs-up with your right hand, and hold it in front of yourself
- 2.
- Pull out your index finger and form a “pistol”. Aim your index finger/ pistol along the first vector $\overrightarrow{a}$.
- 3.
- Pull out your middle finger so that it points straight out from your palm. Twist your hand such that the middle finger points along the second vector $\overrightarrow{b}$. (This is going to look awkward—but take it easy)
- 4.
- Your thumb is now pointing along $\overrightarrow{a}\times \overrightarrow{b}$.

You can use the right-hand rule to explain why the order actually matters in the cross product, because

$$\overrightarrow{a}\times \overrightarrow{b}=-\overrightarrow{b}\times \overrightarrow{a}.$$ |

This is because switching the order of $\overrightarrow{a}$ and $\overrightarrow{b}$ in the product is the same as switching the index finger with the middle finger. To do this, you have to turn your hand upside down, and then your thumb will point in the opposite direction, which gets you the negative of the original cross product.

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