Question Video: Finding the Dot Product of a Standard Unit Vector and Itself Mathematics

Video Transcript

For the unit vectors 𝐢, 𝐣, 𝐤, what is the dot product between 𝐣 and 𝐣?

In this question, we’re told to calculate the dot product between 𝐣 and 𝐣, where 𝐣 is one of our unit directional vectors. And in fact, there’s a few different ways we could evaluate this. The easiest way is to recall something interesting about the dot product. For any vector 𝐯, the dot product between 𝐯 and itself will be the magnitude of 𝐯 squared. And in fact, we know how to prove this. We would do this by writing 𝐯 out component-wise so that we can calculate the dot product between 𝐯 and 𝐯. Remember, to calculate the dot product between 𝐯 and 𝐯, we multiply the corresponding components together and add the results. And we would then see that this gives us the sum of the squares of the components of 𝐯, which is exactly equal to the magnitude of 𝐯 squared.

However, this is a useful result that’s worth committing to memory. Therefore, by using this result with our vector 𝐣, the dot product between 𝐣 and 𝐣 should be equal to the magnitude of 𝐣 squared. And remember, 𝐣 is a unit directional vector. We call it a unit vector because its magnitude is equal to one. We can then use this to evaluate our expression. On the right-hand side of our equation, the magnitude of 𝐣 is equal to one. So this simplifies to give us one squared, which is of course just equal to one. Therefore, we were able to show the dot product between 𝐣 and itself is equal to one.

However, this is not the only way we could evaluate this expression. We could also do it directly from the definition of a dot product. Although it’s not necessary to do this, we’re going to start by writing our vectors 𝐣 out component-wise. Remember, in this scenario, our components are going to be the coefficients of our unit directional vectors 𝐢, 𝐣, and 𝐤. And of course, for 𝐣, the coefficients of 𝐢 and 𝐤 is zero and the coefficient of 𝐣 is one. So our vector 𝐣 is just the vector zero, one, zero. Now we can calculate the dot product between these two vectors directly.

Remember, to calculate the dot product between two vectors, we need to multiply the corresponding components and then add the results. So we start by multiplying the first component of the first vector with the first component of the second vector. This gives us zero times zero. We then multiply the second component of each vector together, giving us one times one. Finally, we multiply the third component of each vector together. This gives us zero times zero. And then we just add all three of these together. And once again, if we calculate this expression, we also see it’s equal to one.

Therefore, we were able to see two different ways of calculating the dot product between the unit directional vector 𝐣 with itself. In both cases, we were able to show this was equal to one.