Question

Determine whether the vectors u and v are parallel, orthogonal, or neither.

u = <8, 4>, v = <10, 7>

pleasssseeeee :)

Answer 1

How do you tell if two vectors are
- parallel
- orthogonal
?

Answer 2

Well, two vectors are parallel if they are in the same direction or opposite directions. Hence just a scalar multiple of each other.

That is, u and v are parallel if there is a number k such that
u = kv

Is there such a number k in this case?

Answer 3

waiting for you to respond. I'm not going to say any more until you do

Answer 4

sorry i was doing other parts of my test..
how do i know theres a number such as k?

Answer 5

The question is this: IS there such a k? Well, if there were then
u = kv
<8, 4> = k<10, 7>
i.e.,
8 = 10k
and
4 = 7k

Can any ONE value of k satisfy both of those equations?

Answer 6

no? lol i dont think so

Answer 7

No indeed. Hence the vectors are not parallel.

Now, what's the test for whether two vectors are orthogonal?

Answer 8

i know that is if they're equal to zero...which theyre not so is it neither??

Answer 9

No. Two vectors are orthogonal if their dot product is zero.

Answer 10

right which i already tested and it isnt correct?

Answer 11

Right. The dot product of u and v is
u.v = 80 + 28 = 108
which is not zero.

Hence the vectors u and v are not orthogonal.

Answer 12

no neither! yay thanks a ton!!
do you mind helping me with two more? lol

Answer 13

Post new questions and if I have time I will. If they are questions about vectors being parallel or orthogonal, try and apply exactly the procedures we used here.

Answer 14

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

u = <-5, 8>, v = <-4, 8>

Answer 15

By the properties of the dot product
\[ u \cdot v = |u| \ |v| \ \cos \theta \]
where \( \theta \) is the angle between u and v.

So use this relationship and rearranging we have
\[ \cos \theta = \frac{ u \cdot v } { |u| \ |v| } \]

From this you can find the angle \( \theta \)

Answer 16

okay so if that value is 68.. where do i go from there

Answer 17

You have a text book or class notes yes? You definitely have worked examples of this sort of "angle between two vectors" problem. Have a look at that.

You have found u.v, which is a start.

Now you need to calculate the length of both vectors, |u| and |v|. And then calculate the value of
u.v/(|u| |v|)

Then you take the inverse cosine of that number.

Answer 18

No, where did you get that?
u = <-5, 8>, v = <-4, 8>

Hence the length of u is
\[ |u| = \sqrt{(-5)^2 + 8^2} = \sqrt{25 + 64} = \sqrt{89} \]

Now do the same to find the length of v, |v|.

Answer 19

oops that value is 84 not 68..

Answer 20

are the lengths just u and v?

Answer 21

By the way, u.v = 20 + 64 = 84 ... yes.

I just showed you a length calculation. It is applying Pythagorus's theorem.

Answer 22

\[\sqrt{80}\]

Answer 23

Yes that's |v|.

Now calculate u.v/(|u| |v|)
because as we saw above, that is equal to the cosine of the angle between u and v.

Answer 24

okay so 84/84.38

Answer 25

ah got it! 5.4 degrees

Answer 26

Sounds about right

Answer 27

ok. I'm out of here. Good luck