Dot products

The dot product is a fundamental way we can combine two vectors. Intuitively, it tells us something about how much two vectors point in the same direction.

We write the dot product with a little dot $\cdot$‍ between the two vectors (pronounced "a dot b"):

If we break this down factor by factor, the first two are $\Vert \overrightarrow{a}\Vert$‍ and $\Vert \overrightarrow{b}\Vert$‍. These are the magnitudes of $\overrightarrow{a}$‍ and $\overrightarrow{b}$‍, so the dot product takes into account how long vectors are. The final factor is $\cos (\theta )$‍, where $\theta$‍ is the angle between $\overrightarrow{a}$‍ and $\overrightarrow{b}$‍. This tells us the dot product has to do with direction.

Specifically, when $\theta =0$‍, the two vectors point in exactly the same direction. Not accounting for vector magnitudes, this is when the dot product is at its largest, because $\cos (0)=1$‍. In general, the more two vectors point in the same direction, the bigger the dot product between them will be.

When $\theta ={\displaystyle \frac{\pi }{2}}$‍, the two vectors are precisely perpendicular to each other. This corresponds to the dot product between them being $0$‍, because $\cos \left({\displaystyle \frac{\pi }{2}}\right)=0$‍.

It's also possible for a dot product to be negative if the two vectors are pointing in opposite directions, which is when ${\displaystyle \frac{\pi }{2}}<\theta <{\displaystyle \frac{3\pi }{2}}$‍.

Another way to think about $\theta$‍ is to imagine one vector dropping a shadow on to the other. When the angle is small, the shadow lands far from the origin and the dot product is large.

When $\theta$‍ is close to ${\displaystyle \frac{\pi }{2}}$‍, the shadow lands close to the origin and the dot product is small.

Keep in mind that the dot product of two vectors is a number, not a vector. That means, for example, that it doesn't make sense to ask what $\overrightarrow{a}\cdot \overrightarrow{b}\cdot \overrightarrow{c}$‍ equals. Once we evaluated $\overrightarrow{a}\cdot \overrightarrow{b}$‍ to be some number, we would end up trying to take the dot product between a number and a vector, which isn't how the dot product works.

When we need to find a dot product in multivariable calculus, we typically have only the coordinates of $\overrightarrow{a}$‍ and $\overrightarrow{b}$‍. Calculating $\Vert \overrightarrow{a}\Vert \Vert \overrightarrow{b}\Vert \cos (\theta )$‍ would force us to find two square roots and a cosine, which is a lot of work! Luckily, there is an easier way. Just multiply corresponding components and then add:

Although the example above features 3D vectors, this formula extends for vectors of any length.

This makes calculating dot products straightforward if you know the components of each vector.

Let's try an example.

Even though we now have a nice formula in terms of coordinates, the intuition behind the dot product remains how it measures relative direction. Try to predict the sign of the dot product based on just a picture.

That's all we'll need to know about dot products for now. If you want to learn more, check out this video.

Now that we've covered the dot product, there's just one more vector operation to learn about: the cross product. As we'll see, the cross product nicely complements the dot product but is slightly more limited.