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The Angle Between Vectors - Problem 2
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
When are two vectors parallel, and when are they perpendicular? Let’s start with parallel, it’s the easier case. U is parallel to v if and only if, u is a scalar multiple of v. If there’s some constant k, not zero, that you can multiply by v and get u, then the two vectors are parallel. That’s true even if the vectors are in opposite directions, we still consider them parallel.
What about perpendicular? Well it boils down to our result that finds the angle between two vectors. Cosine theta equals u.v over the magnitude of u times the magnitude of v. This angle is going to be 90 degrees whenever this number is zero. Because remember, the cosine of 90 degrees is zero. I need the dot product to be zero. That’s the only time that two vectors are going to be perpendicular. So u is perpendicular to v if and only if, u.v is zero. It’s a nice easy quick test for perpendicularity. That's work.
Let’s take a look at a problem; Prove that ABCD is a rectangle. Here I’m going to use a theorem from Geometry that says, if ABCD is a parallelogram and it has one right angle, then it’s a rectangle. Let’s first show it’s a parallelogram. Just a quick sketch, ABCD, doesn’t have to be accurate. I just want to show you that when you label any kind of quadrilateral you label it so that these letters are consecutive vertices. So ABCD tells me how the vertices are laid out.
Let’s first find the vector AB. Vector AB in components, will be 2 minus -4, 6, 17 minus 9 which is 8. BC, the second side, is going to be 14 minus 2. Remember this picture is not accurate. 8 minus 17 which is -9. CD; 8 minus 14, -6, 0 minus 8, -8 and then DA. This is going to be -4 minus 8, -12, 9 minus 0, 9.
Let’s make an observation, AB and CD are opposite sides, and AB equals CD times -1. AB is -1 times CD. This is a constant. This means that AB and CD are parallel. AB is parallel to CD. What about the other two sides? Well, BC is -1 times DA. So those two are parallel as well. And that proves, since both pairs of opposite sides are parallel, that proves it’s a parallelogram.
Is it a rectangle? All I have to do is show that one angle is a right angle. I do that by considering, I’ll take any of these. How about AB.BC. If I can show that this dot product is zero, then I’ll know that this angle is a right angle and it will be a rectangle. AB.BC; AB is <6, 8> , BC is <12, -9> I get 6 times 12, 72, plus 8 times -9, plus -72, perfect, zero. And that proves it. That proves that angle B, this angle, but this drawing is not accurate, but this angle is now a right angle in a parallelogram, therefore ABCD is a rectangle.
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