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Addition and Scalar Multiplication of Vectors - Problem 2
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Recall what the opposite of a number is; for example the opposite of 5 is -5. It's just -1 times that number. Now that we have scalar multiplication, we can do the same thing with vectors. We can talk about the opposite of a vector u, and we define it as the opposite of vector u is scalar -1 times u. So for example if u is -9,4 then the opposite of that would be +9,-4, you just multiply -1 by each of the components.

Now, once you have the idea of an opposite, we can define vector subtraction. U minus v is defined as u plus the opposite of v, let me make that look more like a v. So if I have the components of u as u1,u2 and v as v1,v2, then u minus v becomes u1 minus v1, u2 minus v2. So let's try this out in a problem.

The problem says; u equal 5,3 and vector v is -2,1. Compute 2u minus v. So first I'm going to calculate 2u, the scalar of multiple 2 times u. And that's going to be 2 times 5 or 10 and 2 times 3 or 6, minus vector v -2,1. And our rule for subtraction tells us that we subtract component-wise. So this is going to be 10 minus -2, or 10 plus 2, 12, and 6 minus 1, 5 so 12,5.

Now notice here I'm calculating v minus 2u, a very similar difference, but it's the opposite difference, so let's see what it gives us. We have v which is -2, 1 minus 2u, which I already calculated is 10,6. So when I subtract again I subtract component-wise, -2 minus 10 is -12 and 1 minus 6, -5. So it probably won't surprise you that v minus 2u is literally the opposite of 2u minus v.

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