Question

Okay, so i've been stuck on this problem for a week now and every time i try and work it out, i confuse myself more and more.. please help :/

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

u = <8, 7>, v = <9, 7>

-8.3°
1.7°
3.3°
13.3°

Answer 1

Have you learned the dot product
\[ \vec{u} \cdot \vec{b} = | \vec{u}| | \vec{v}| cos \theta \]

Answer 2

Im learning it online but struggling trying to teach it to myself

Answer 3

Derp! Wrong letter. Shouldn't be a b in the first one.

\[ \vec{u} \cdot \vec{v} = | \vec{u}| | \vec{v}| cos \theta \]

Okay, so for the dot product, by definition we have

\[ \vec{u} = <u_1, u_2>\]
and
\[ \vec{v} = <v_1, v_2>\]

\[\vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2\]

Answer 4

The magnitude of one of the vectors is
\[| \vec{u} | = \sqrt{u_1^2 + u_2^2}\]

Answer 5

Yeah thats as far as i understand.. i got 129 and then i got confused because then i see the degre sign lol

Answer 6

And so to find the angle, it would simply be \[\theta = \cos^{-1}\left(\frac{\vec u \cdot \vec v}{|\vec u||\vec v|}\right)\]

Answer 7

And don't even think too hard about it, just ask yourself, what is \(\theta\) measured in? Degrees or radians. Pick your poison.

Answer 8

degrees right?

Answer 9

Mmhmm, that's what your answer choices are.

Answer 10

Can you explain how i could go about doing it? .. sorry its been a while since ive learned anything that has to do with math .. and im really rusty

Answer 11

First find \(\vec u \cdot \vec v\) as @AllTehMaffs described above.

Answer 12

Then find the magnitude of each, \(|\vec u| = \sqrt{u_1^2+u_2^2}\) and \(|\vec v|=\sqrt{v_1^2 +v_2^2}\)

Answer 13

Finding those just plug them into the formula I have written,

\(\color{blue}{\text{Originally Posted by}}\) @Jhannybean
And so to find the angle, it would simply be \[\theta = \cos^{-1}\left(\frac{\vec u \cdot \vec v}{|\vec u||\vec v|}\right)\]
\(\color{blue}{\text{End of Quote}}\)

And just plug it into your calculator and you'll find your answer :)

Answer 14

(8)(9)+(7)(7)
72+ 49
129
-------------------
so for the magnitude part i got u = 10.63 and v= 11.40 ... right or wrong?

Answer 15

\[\theta =\cos^{-1} \left[\begin{matrix}(8)(9) + (7)(7) \\ \left| 10.63 \right| \left| 11.40 \right|\end{matrix}\right]\]

Answer 16

could I suggest you draw it and just use pythagoras
see drawing just another way of thinking of it

Answer 17

So the answer is C. 3.3 ?