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The Cross Product of Vectors - Problem 2
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
I want to prove a really important property of the cross product. So we have vectors u and v from a previous example I want to show that they're cross products. u cross v is perpendicular to both u and v. This is a really important property. We actually calculated u cross v in a previous example. This was our answer; i minus 4j plus 5k.
And the way I’m going to show that u cross v is perpendicular to u or v, is I’m going to take the dot product of the two vectors. I have to do both of these. I have to show u cross v is perpendicular to u. So I’ll start by taking the dot product of u cross v and u, then I’ll put those into component form.
U cross v is <1, -4, 5> and u is <4, 1, 0>. And so the dot product is going to be 4 minus 4 plus 0, which is 0. Therefore u cross v is perpendicular to u, so that’s what we’ve just shown. Remember if the dot product to two vectors is 0, the two vectors are perpendicular.
And then let’s take u cross v dot v. You cross v is again <1, -4, 5> and v is <-5, 0, 1>. So the dot product is -5 plus 0 plus 5 also 0. So u cross v is perpendicular to v.
So this is a really important property about the cross product. First of all the cross product is the kind of multiplication. And one of the things you probably notice about vectors is, multiplication is strictly business. We have actually three kinds of product. We have the scalar product of a vector, we have the dot product and we have the cross product.
Now the cross product is the only product of vectors that results in a vector. This is a vector and it's a product of these two vectors. And it’s important that you know that what kind of vector you get is perpendicular to the original two, not always happens with the cross products.
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