Video: Calculating the Vector Product of Two Vectors

Consider the two vectors 𝐑 = 3𝐒 + 2𝐣 and 𝐒 = 5𝐒 + 8𝐣. Calculate 𝐑 Γ— 𝐒.


Video Transcript

Consider the two vectors 𝐑 equals three 𝐒 plus two 𝐣 and 𝐒 equals five 𝐒 plus eight 𝐣. Calculate 𝐑 cross 𝐒.

Well, we see this exercise is about computing a vector product between these two vectors 𝐑 and 𝐒. We’ve been given those two vectors in component form. And we see that each one has an 𝐒-component, that is a component along the π‘₯-axis, as well as a 𝐣-component, some component along the 𝑦-dimension. We see then that both of these vectors, 𝐑 and 𝐒, lie in the π‘₯𝑦-plane. In fact, if we were to sketch out π‘₯- and 𝑦-axes, then we can draw in vectors 𝐑 and 𝐒 on this graph.

Vector 𝐑 extends three units in the positive π‘₯-direction and two units in the positive 𝑦, giving a vector like this. While vector 𝐒 extends five units in the positive π‘₯-direction and then eight units in the positive 𝑦-direction. And that vector looks like this. Now, to calculate this vector product, 𝐑 across 𝐒, we’ll want to recall the mathematical form of the vector product of two vectors that lie entirely in the π‘₯𝑦-plane like 𝐑 and 𝐒 do.

If we call two general vectors that both are constrained to lie in the π‘₯𝑦-plane 𝐀 and 𝐁. Then the vector product of 𝐀 and 𝐁, also called the cross product of 𝐀 and 𝐁, is equal to the π‘₯-component of 𝐀 times the 𝑦-component of 𝐁 minus the 𝑦-component of 𝐀 times the π‘₯-component of 𝐁. Note that a vector product of two vectors results in a vector. That is, it has both magnitude as well as direction. And since our vectors 𝐀 and 𝐁 both lie in the π‘₯𝑦-plane, their vector product points perpendicularly to that, in the 𝐀-direction.

Let’s now use this relationship along with 𝐑 and 𝐒 written in their component form to calculate 𝐑 cross 𝐒. This vector product is equal to the π‘₯-component of 𝐑 times the 𝑦-component of 𝐒 minus the 𝑦-component of 𝐑 times the π‘₯-component of 𝐒. And this will also point in the 𝐀-direction, either positive or negative. Looking at these different terms in our parentheses, we can see that the π‘₯-component of 𝐑 is three, that the 𝑦-component of 𝐒 is eight and then the 𝑦-component of 𝐑 is two and the π‘₯-component of 𝐒 is five.

When we substitute in these values, our next task is to calculate their result. Three times eight is 24, and two times five is 10. And then, 24 minus 10 is equal to 14. So, our vector product points 14 units in the positive 𝐀-direction. So, if we drew in a 𝑧-axis on our graph, where that axis pointed out of the screen at us, then this vector product would point 14 units toward us along that direction. This is the vector product of 𝐑 and 𝐒.

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