Video Transcript
The diagram shows two vectors, 𝐀 and 𝐁. Calculate the magnitude of the vector product of 𝐀 and 𝐁. Give your answer to the nearest integer.
The diagram shows us that the magnitude of the vector 𝐀 is 12, the magnitude of the vector 𝐁 is 16, and the angle between them is 82 degrees. We are asked to calculate the magnitude of the vector product of these two vectors. The vector product is sometimes also known as the cross product. There is actually a simple formula for this magnitude in terms of the magnitude of the two vectors and the angle between them, which is exactly the information we have from the diagram.
The formula is that the magnitude of the vector product of 𝐀 and 𝐁 is the magnitude of 𝐀 times the magnitude of 𝐁 times the sin of 𝜃, where 𝜃 is the angle between the two vectors, which in this case is 82 degrees, as marked on the diagram. Note that we could also think about this angle here between the two vectors, which is 360 degrees minus 82 degrees. That is the complement of 82 degrees needed to make a full circle. Note that in order to make sure that the magnitude is always greater than or equal to zero, we need 𝜃 to always be between zero and 180 degrees. So when we have the choice to either use 82 degrees or 278 degrees, we choose 82 degrees because that way the magnitude will be positive.
It’s worth mentioning that using the larger angle instead of the smaller angle only changes the answer by a negative sign. So another way to ensure that 𝐀 cross 𝐁 is always positive or zero is to use whichever angle we choose and then take the absolute value of the right-hand side. In either case, the way the problem is given to us, 𝜃 is in fact the smaller angle. So we’ll just use that and not worry about absolute values.
Using the values from the diagram, the magnitude of 𝐀 cross 𝐁 is 12 times 16 times the sin of 82 degrees, which is equal to 190.131 and several more decimal places. We are told to give our answer to the nearest integer. And 190.131 et cetera rounded to the nearest integer is 190. And this is, to the nearest integer, the magnitude of the vector product of the two vectors 𝐀 and 𝐁 shown in the diagram.
