Vectors and Parametric Equations
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
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.