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Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.)

$ a = \langle 4, 3 \rangle , b = \langle 2, -1 \rangle $

$$\cos ^{-1}\left(\frac{1}{\sqrt{5}}\right)=63^{\circ}$$

01:07

Alan G.

Calculus 3

Chapter 12

Vectors and the Geometry of Space

Section 3

The Dot Product

Vectors

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for this problem. We have been given two vectors A and B, and we want to find the angle between these two vectors. We're going to find it first, is an exact number and then approximated to the nearest degree. Now we have a the're, um that tells us how we how the angle between two vectors is related to the vectors themselves. Our formula says that the co sign of fate a fate of being the angle between A and B the co sign of that angle is the dot product of our two vectors over the multiplication of the magnitude of those two vectors. Okay, so we dot product on top, and then we multiply the magnitudes on the bottom. So let's see what we have. Let's find our pieces first of all dot product For the dot product, we multiply the X co uh, the X pieces together, and then we add the multiplication of the white pieces. So for this particular case, are dot product is going to be eight plus negative three or five. Now, let's look at the magnitudes over magnitude of we're just finding the distance that the that distance along my vector so that's going to be using, um, kind of kind things. This Pythagorean theorem. Okay, I've got four squared plus three squared that's gonna give me 25. So the magnitude of Vector A is five. How about the magnitude of Vector be against square root? We're gonna take our first coordinate squared, plus our second coordinate squared. That's going to give me the square root of five. So now I have all of the pieces. Let's put this into our formula. Co sign of theta equals the dot product, which in this case is five over the magnitude of the first times the magnitude of the second. Those fives will cancel. What we're left with is co sign of Fada equals one over the square root of five. So if I want to know, what they did is I'm just trying to take in verse co side of one over square root of five. That's my exact answer for this angle. Plugging that into a calculator to get the angle to the nearest degree. It gives us an angle of approximately 63 degrees. It's a little bit bigger, but it rounds to the nearest degree is 63. So here's my angle between my given vectors

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