Question

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

-9.1°
1.8°
0.9°
11.8°

Answer 1

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

Answer 2

\[\Large\vec u\cdot\vec v=\|\vec u\|\|\vec v\|\cos\theta_{uv}\]

Answer 3

\[\Large\theta_{uv}=\cos^{-1}\left({\vec u\cdot\vec v\over\|\vec u\|\|\vec v\|}\right)\]

Answer 4

I'm still not too sure where to go with this problem.
I didn't understand my teacher during the lesson.

Answer 5

do you know how to find the length of a vector?
i.e. can you get \(\|\vec v\|\) from \(\vec v\) ?

Answer 6

not really, i was absent because of a winter break trip. I missed the lesson, and I didn't understand my teacher in the review lesson.

Answer 7

let's draw a vector \(\vec v\) with components \(\langle a,b\rangle\)

Answer 8

now we know the components are <a,b>, so the horizontal length is \(a\) and the vertical length is \(b\)now do you have an idea how to find the length of \(\vec v\) ?

Answer 9

c^2=a^2+b^2

Answer 10

perfect, so the length of \(\vec v\) is what we call the "magnitude" of \(\vec v\), written as \(\|\vec v\|\)
so now we know that for any vector \(\vec v=\langle a,b\rangle\), we have that\[\|\vec v\|=\sqrt{a^2+b^2}\]

Answer 11

so now you know how to find the \(\|\vec u\|\|\vec v\|\) part, right?

Answer 12

so what would a and b be?

Answer 13

the components of the vector

Answer 14

you are given u = <-5, -4>
you should be able to tell me the components from this

Answer 15

-5.-4?

Answer 16

yes, for \(\vec u\)

Answer 17

oh okay thats what i thought

Answer 18

coolness, now what about the dot product?
I am guessing you also do not know how to find the dot product of two vectors \(\vec u\cdot\vec v\), right?

Answer 19

nope

Answer 20

my teacher isn't the best :(

Answer 21

it happens :P
there are two ways. If you know the angle between the two angles, you can use the formula I wrote at the beginning\[\Large\vec u\cdot\vec v=\|\vec u\|\|\vec v\|\cos\theta_{uv}\]but in your problem we are trying to find \(\theta\), so we need to use the other formula for the dot product first, which is...

Answer 22

if\[\vec v=\langle a,b\rangle\]\[\vec u=\langle x,y\rangle\]then\[\vec u\cdot\vec v=ax+by\]

Answer 23

so what is \(\vec u\cdot\vec v\) for the vectors you are given in your problem?

Answer 24

so 20+12?

Answer 25

32?

Answer 26

yep :)

Answer 27

so nowyou can use this formula:\[\vec u\cdot\vec v=\|\vec u\|\|\vec v\|\cos\theta_{uv}\]and solve for \(\theta\)
(make sure to check that your answer is in the right quadrant at the end)

Answer 28

actually i guess quadrant doesn't matter here, since you are looking for the angle between them. You can just blindly use the formula.

Answer 29

what do you mean blindly use it?

Answer 30

I mean just plug in to the formula, and the number that comes out is your answer
sometimes in trig you need to add 180 degrees, but not in this case

Answer 31

what do i do to find the absolute value of vector u and v

Answer 32

is it 1.8?