Video Transcript
๐ด๐ต๐ถ๐ท is a square of side four, and ๐ฎ is a unit vector perpendicular to the squareโs plane. Find ๐๐ cross ๐๐.
Looking at our square, we see that corners ๐ด and ๐ท are in these spots and ๐ต and ๐ถ are here. Our question asks us to define vectors, one from point ๐ด to point ๐ท and the other from point ๐ต to point ๐ถ, and then cross them. Vector ๐๐ looks like this and ๐๐ like this. Since these vectors are on the sides of this square, we know that both of them have a magnitude of four. They both also point in the same direction in the plane of our screen.
Now, in general, if we have two vectors, weโll call them ๐ and ๐, and we cross them, then thatโs equal to the magnitude of one vector times the magnitude of the other times the sine of the angle between them. Here weโve called that angle ๐, all in the direction of a vector thatโs perpendicular both to vector ๐ and vector ๐.
Applying this relation to our scenario, on the right-hand side, we again use ๐ to specify the angle between our two vectors ๐๐ and ๐๐. This vector ๐ฎ weโre told is a unit vector thatโs perpendicular to the squareโs plane. That is, it either goes into or out of the screen. Now, we know both the magnitude of ๐๐ and the magnitude of ๐๐. Theyโre both four. But then letโs think about the sine of the angle between these vectors. Since both vectors point in exactly the same direction, the angle between them is zero degrees.
We can write then that this cross product equals four times four times the sin of zero degrees in the ๐ฎ-direction. But then, the sin of zero degrees is itself equal to zero. And that means our entire expression, this cross product, is zero. Whenever two vectors are parallel, as they are in this case, or antiparallel, then the cross product between them must be zero. Thatโs because, for parallel vectors, the sin of zero degrees is zero. And for antiparallel vectors, the sin of 180 degrees is zero. ๐๐ cross ๐๐ then equals zero.
