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Addition and Scalar Multiplication of Vectors - Problem 1

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

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.

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