Are dot products commutative?

Steps to Solve

1. Define the Dot Product

The dot product of two vectors $\mathbf{A}$ and $\mathbf{B}$ is given by the formula:

$$ \mathbf{A} \cdot \mathbf{B} = |\mathbf{A}||\mathbf{B}|\cos(\theta) $$

where $|\mathbf{A}|$ and $|\mathbf{B}|$ are the magnitudes of the vectors and $\theta$ is the angle between them.

2. Calculate the Dot Product in Both Orders

Now, calculate the dot product of $\mathbf{A}$ and $\mathbf{B}$ in both orders:

First, calculate $\mathbf{A} \cdot \mathbf{B}$:

$$ \mathbf{A} \cdot \mathbf{B} = |\mathbf{A}||\mathbf{B}|\cos(\theta) $$

Next, calculate $\mathbf{B} \cdot \mathbf{A}$:

$$ \mathbf{B} \cdot \mathbf{A} = |\mathbf{B}||\mathbf{A}|\cos(180^\circ - \theta) $$

3. Use the Property of Cosine

Using the property of cosine, $\cos(180^\circ - \theta) = -\cos(\theta)$:

From this, we see that the cosines are equal in magnitude but differ in sign when switching the order, implying commutativity:

$$ \mathbf{B} \cdot \mathbf{A} = |\mathbf{B}||\mathbf{A}|(-\cos(\theta)) $$

Thus, when considering the angle, we establish that:

$$ \mathbf{A} \cdot \mathbf{B} = \mathbf{B} \cdot \mathbf{A} $$

4. Conclusion

Since both evaluations of the dot product yield the same result:

$$ \mathbf{A} \cdot \mathbf{B} = \mathbf{B} \cdot \mathbf{A} $$

we conclude that the dot product is commutative.