Video Transcript
For the unit vectors 𝐢, 𝐣, and 𝐤, what is the dot product between 𝐢 and 𝐣?
In this question, we want to calculate the dot product between vectors 𝐢 and 𝐣 where 𝐢 and 𝐣 are our standard unit directional vectors. There’s actually several different ways we can evaluate this. We’ll go through a few of these. The first way we could do this is to recall a formula involving the dot product and the angle between two vectors. We recall if 𝜃 is the angle between two vectors 𝐮 and 𝐯, then the cos of 𝜃 will be equal to the dot product between 𝐮 and 𝐯 divided by the magnitude of 𝐮 times the magnitude of 𝐯. And this gives us a useful result for calculating the dot product between 𝐮 and 𝐯 if we know the magnitude of 𝐮, the magnitude of 𝐯, and the angle between these two vectors.
And in this case, we know all three of these values. First, recall that our vectors are the unit directional vectors. We call them unit vectors because their magnitude is equal to one. And it’s worth pointing out we represent this on our vectors with the hat notation. But remember, the unit directional vectors are perpendicular to each other. And if two vectors are perpendicular to each other, then the angle between these two vectors must be equal to 90 degrees. We can then use this equation to evaluate the dot product between 𝐢 and 𝐣.
First, because we know the angle between 𝐢 and 𝐣 is 90 degrees, we must have the cos of 90 degrees is equal to the dot product between 𝐢 and 𝐣 divided by the magnitude of 𝐢 times the magnitude of 𝐣. And as we’ve already discussed, 𝐢 and 𝐣 are unit vectors, so their magnitudes are just equal to one. Therefore, this entire expression just simplifies to give us the dot product between 𝐢 and 𝐣 is equal to the cos of 90 degrees. And the cos of 90 degrees is just equal to zero. Therefore, the dot product between 𝐢 and 𝐣 is just equal to zero.
However, this is not the only way we could have calculated this dot product. We can also do this directly from the definition of a dot product. And although it’s not necessary, we’re going to start by writing our vectors 𝐢 and 𝐣 component-wise. Remember, the components of our vectors represent the coefficients of 𝐢, 𝐣, and 𝐤. For vector 𝐢, the only nonzero coefficient is the coefficient of 𝐢, which is equal to one. And for 𝐣, the only nonzero coefficient is the coefficient of 𝐣, which is also equal to one. So 𝐢 is the vector one, zero, zero, and 𝐣 is the vector zero, one, zero.
And it’s also worth pointing out we could’ve just ignored the third component completely. It won’t change our answer, and both methods are correct anyway. The only reason we’re including the third component is because the question specifically asked us to include all three direction vectors. So we leave this as three components; however, it is personal preference if you prefer not to.
Now, we want to find the dot product between these two vectors. And remember, to do this, we want to multiply the corresponding components and then add all of these products together. So we start by multiplying the first components of our two vectors together. This gives us one multiplied by zero. Then we add on the product of the second components of our vectors. That’s zero multiplied by one. And finally, we add the product of the third components of our vectors. That’s zero multiplied by zero. And we can calculate this. All three of these terms have a factor of zero, so they’re all just equal to zero.
Therefore, we’ve shown in two different ways the dot product between the unit directional vectors 𝐢 and 𝐣 is equal to zero.
