Question

Find the angle between the given vectors to the nearest tenth of a degree.

u = <-5, -4>, v = <-4, -3>

Answer 1

So the formula is:
\[\cos \theta = \frac{ u*v }{ ||u||*||v|| }\] So this means multiply vectors u and v in the numerator and multiply their magnitudes in the denominator. Do you kno whow to perform either of those operations?

Answer 2

i did but i can't remember what to do for the denominator

Answer 3

Ah, okay. Well the magnitude is found by doing the pythagorean theorem with i and j. So for vector u, i is -5 and j is -4. So pythagorean theorem with this gives me:
\[\sqrt{(-5)^{2}+(-4)^{2}} = \sqrt{41}\] See what i did?

Answer 4

yeah!

Answer 5

Alright, cool. Now we do that for vector v, which is -4, -3
\[\sqrt{(-4)^{2}+(-3)^{2}} = 5\]
So now we have:
\[5\sqrt{41}\] in the denominator. You remember how to multiply vectors with dot product?

Answer 6

i'm following you but i'm not sure what to do next

Answer 7

Well, I just did the denominator part since that was you said you weren't sure what to do with. So we're not quite sure how to do the numeraotr?

Answer 8

is the top 240?

Answer 9

Way too large.

Answer 10

So if we have two vectors u and v, each ofthe form of i + j. Then uv = the multiplication of the i's + the multiplication of the j's.
\[u = u _{1}i + u _{2}j\]
\[v = v _{1}i+ v _{2}j\]
\[uv = (u _{1})(v _{1}) + (u _{2})(v _{2})\]
If that makes any sense.

Answer 11

so -16?

Answer 12

(-5)(-4) + (-4)(-3)
Check again :P

Answer 13

oh wow i'm losing it haha 32

Answer 14

so now what do i do

Answer 15

Checking, lol. I'm not getting something pretty xD

Answer 16

oh boyyyyy hahah

Answer 17

Put it in my calculator wrong xDD

Answer 18

these are the answer choices :)
-9.1°

1.8°

0.9°

11.8°

Answer 19

Yeah, I found the answer, I just put it in the calculator wrong. Okay, so we have the result of all our multiplying. So now we have this:
\[\cos \theta = \frac{ 32 }{ 5\sqrt{41} }\]
Now just need to solve for theta :P

Answer 20

0.9?

Answer 21

I got 1.8.

Answer 22

i got 0.9999 but that would round up :/ i might be plugging it in wrong

Answer 23

Yes, you should get that (don't round). But that is what cos(theta) is equal to. But wedon't want cos(theta), we want the actual theta. Do you know how to get that from here?

Answer 24

noo :/

Answer 25

Well, right now we have:
\[\cos \theta = number\]But we already know the number, we want theta. Well when we want theta, we do this:
\[\cos^{-1} (number) = \theta \]

Answer 26

GOT IT thank you so much

Answer 27

Okay, awesome ^_^