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Hello, everybody.
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In this video, i'm going to be showing you how to solve exercise 75 in chapter 13, section 4, of calculus early transcendentals.
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Now, in this problem, we are given two non -zero vectors, u and v, and they want us to prove that the vector equation u cross z equals v can only be solved for the vector z if and only if the dot product between u and v is zero.
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Now, to do this, what we want to do is take this expression here, and we want to do.
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You with both sides of the equation.
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On the left -hand side, this gives us u dotted with u cross z.
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Now, if we want to manipulate the left -hand side of this equation, what we can do is use a result from exercise 72.
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So if you don't understand this next step, i would recommend you go back to this problem and work out for yourself that this property is true.
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But what we can do is cyclically rearrange these three terms in this quantity here and pull z out front and have it dotted with u cross u.
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Now, what we want to notice here is that this cross product, u cross u is a cross product between the same vector twice.
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So this is going to be equal to the zero vector...