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Addition and Scalar Multiplication of Vectors - Problem 1
Let's do a problem that involves vector addition and scalar multiplication. If the vector u equals 5,-2 and vector v is 3 and 4. Compute u plus v. Now remember that we add vectors component-wise, so u is 5,-2 plus v is 3,4. So the sum of these two vectors is going to be 5 plus 3, 8 and -2 plus 4, 2, so that's the sum.
Now here I've got a combination of adding vectors and scalar multiples. I'm going to calculate the scalar multiples first. So -2 times u, I'm going to take this -2, and distribute it over the components of u. So -2 times 5 is -10 and -2 times -2, +4, so that's -2 times the vector u. Now plus 3v I want to calculate 3v and put it here, that's 3 times 3 which is 9 and 3 times 4 which is 12. So I add these guys, component-wise and I get -1,16. Minus 1 times the vector u plus v, so this is a scalar multiple, I'll calculate that first.
Minus 1 times vector u is -5, +2. It just reverses the signs of the two components, and then v I'll just write down v, 3,4 and then I add component-wise, so -5 plus 3, -2, 2 plus 4, 6. And just so we don't forget how to do, notice in part d I ask you to find the magnitude of -1 times u plus v. This is exactly the vector we just found. So we're looking for the magnitude of -2, 6 and recall that that's the square root of the sum of the squares of the components. So -2² plus 6². So that's 4, plus 36, 40, this is root 40 and root 40 has a factor of 4, so this becomes 2 root 10, that's it.
Once you have the idea of the components, addition and scalar multiplication is really easy, all you have to do is for addition, add vectors component-wise, and for scalar multiplication, multiply the scalar through to each of the components.