# Ex 2: Properties of Cross Products – Cross Product of a Sum and Difference

Here we’re given the components of vector u crossed with vector v, and we’re out to find the cross product of vector u minus four times vector v, and vector u plus two times vector v. So because we’re not given vector u and v, we’ll have to find this cross product using the properties of cross products given here below. Let’s focus on this property first, where if we have vector u crossed with the sum of vector v and w, this is equal to vector
u crossed with vector v plus vector u crossed with vector w. This is very similar to
the distributive property in algebra. So looking at our problem, the property doesn’t seem to fit exactly, but if we take a look at vector u minus four times vector v, this is just some resultant vector so if we call this vector
w, if it’s helpful, we can think of this as vector w crossed with the sum of these two vectors. So if we find this with w, this would give us vector
w crossed with vector u and then plus vector w crossed with two times vector v. And again we know vector
w is just vector u minus four times vector v, so let’s go ahead and make
that substitution now. We would have vector u
minus four times vector v crossed with vector u plus, again, vector u minus four times vector v crossed with two times vector v. And now we can apply this property again. Here we have vector u
crossed with vector u minus four times vector
v crossed with vector u, and then we’d have plus u crossed with two times vector v,
minus four times vector v crossed with two times vector v. Now because we know
vector u crosses vector v we want all these cross products to be in this order. And we also know when we cross that vector with itself, we get the zero vector. So notice how this first cross product would be the zero vector. We want to change the
order of this cross product so we have vector u crossed with vector v so notice how that’s
going to change the sign, so this would be positive four times vector u crossed with vector v, and then we can write this cross product as plus two times vector
u crossed with vector v, and here this would simplify to minus eight times vector v crossed with vector v, but notice how this
would be the zero vector. So this simplifies to the zero vector so simplifying one more time this would be six times vector u crossed with vector v. And we know vector u crossed with vector v because that was given, so this is equal to six
times the cross product with an x component of negative one, a y component of three, and a z component of one. So this would give us an x component of negative six, a y component of 18, and a z component of 6. I hope you found this helpful. ## 1 thought on “Ex 2: Properties of Cross Products – Cross Product of a Sum and Difference”

1. Irene Mahoney says:

You switched the sign when you made (-4v x u) into (4u x v) but then didn't when you changed (u x 2v) to (2u x v)?