Video Transcript
Given that vector 𝐀 and vector 𝐁 satisfy 𝐀 cross 𝐁 equals the zero vector, 𝐀 does not equal the zero vector, and 𝐁 does not equal the zero vector, how are the two vectors related? (A) Parallel, (B) perpendicular.
In this example, we’re told that the cross product of these two vectors 𝐀 and 𝐁 equals the zero vector. This is a vector that has a magnitude of zero. So therefore, any direction associated with it has no meaning. We could equivalently write then that the cross product of 𝐀 and 𝐁 equals zero. This is what we’re being told in the statement, and it can bring to mind the general mathematical relationship for the cross product of two vectors. Given vectors 𝐮 and 𝐯, 𝐮 cross 𝐯 is equal to the magnitude of vector 𝐮 times the magnitude of vector 𝐯 times the sine of the angle between the two vectors. Here we’ve called that angle 𝜃. And since the cross product of two vectors is itself a vector, this result has a direction associated with it.
𝑛 hat, we’ll say, is a unit vector that’s in a direction normal or perpendicular to both vectors 𝐮 and 𝐯. Since, in our instance, we have two vectors whose cross product is zero, we can look at this general equation and think of ways that that could be true. One way that 𝐮 cross 𝐯 could be zero is if the magnitude of 𝐮 is zero, or if the magnitude of 𝐯 is zero, or if the sin of 𝜃 is zero. In our example, we’re told that neither vector 𝐀 nor vector 𝐁 has a magnitude of zero. That is, they’re not equal to the zero vector. This means the only way it can be true that 𝐀 cross 𝐁 is equal to zero is if the sine of the angle between them is zero.
If we recall the shape of the sine function over one complete cycle from zero degrees up to 360 degrees, we see that the sin of 𝜃 is zero when 𝜃 itself is zero degrees, or when it’s 180 degrees or 360 degrees, which is the same as zero degrees one cycle later. So then, if our two vectors 𝐀 and 𝐁 are separated either by zero degrees or by 180 degrees, then the sin of the angle between them will be zero, and their cross product will be zero. If 𝐀 and 𝐁 are separated by zero degrees, they would look like this. They would be parallel vectors.
On the other hand, a 180-degree separation would look something like this. These vectors are antiparallel. In either case, 𝐀 cross 𝐁 would be zero. But we see among our answer options that only parallel and not antiparallel is listed. Since this is one of the two ways that vectors 𝐀 and 𝐁 could be related so that their cross product is zero, we’ll choose to say these vectors are parallel to one another.
