Question

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

Answer 1

u=(1,2) v=(-4,8)

Answer 2

I know the answer is not orthogonal because the dot product does not = 0. I need help in determining if it's parallel

Answer 3

Parallel vectors mean the angle between them is \(0^\circ\)

Orthogonal (perpendicular) vectors have an angle between them that = \(90^\circ\).

Answer 4

OK I had it mixed up then. So they are not parallel

Answer 5

So in order to determine this, you need to use the dot product and the cross product of vectors that give you an angle measurement.

Dot Product: \(\vec a \cdot \vec b = |a||b|\cos(\theta)\)

Cross product: \(\vec a \times \vec b = |a||b|\sin(\theta)\)

Answer 6

So u = -4, and v= -16

Answer 7

\[\theta = \cos^{-1}\left(\frac{\vec a \cdot \vec b}{|\vec a||\vec b|}\right) \]\[\theta = \sin^{-1}\left(\frac{\vec a \times \vec b}{|\vec a||\vec b|}\right)\]

Answer 8

and theres a typo in my first post. It should be \[\vec a \cdot \vec b = |\vec a||\vec b|\cos(\theta)\]\[\vec a \times \vec b = |\vec a||\vec b|\sin(\theta)\]

Answer 9

So i'll start you off. \[\vec u = \langle a_1~,~a_2\rangle \implies \vec u = \langle 1~,~2 \rangle\]\[\vec v = \langle b_1~,~b_2 \rangle \implies \vec v =\langle -4~,~8\rangle\] \[\vec u \cdot \vec v = (a_1b_1) +(a_2b_2)\]\[|\vec u| = \sqrt{a_1^2+a_2^2}\]\[|\vec v| = \sqrt{b_1^2+b_2^2}\]

Answer 10

now just solve for all these parts and plug them in to the formula. Depending on whether you get \(0^\circ\) or \(90^\circ \) , you can determine whether its parallel or perpendicular.

Answer 11

The 2 is suppose to be a -2 that was a typo
-4+(-16)= -20

Answer 12

Which part are you solving for?

Answer 13

|u| = 2.23 or sqrt 5
|v|= 8.94 or 4 sqrt 5

Answer 14

\[|\vec u| = \sqrt{1^2+(-2)^2} = \sqrt{5} =2.2~\checkmark \]\[|\vec v| =\sqrt{(-4)^2+8^2} = \sqrt{16+64}= \sqrt{80}=4\sqrt{5} =8.94 ~\checkmark\]

Answer 15

Next step?

Answer 16

Now you should find \[\vec u \cdot \vec v\]\[\vec u \times \vec v\]

Answer 17

I'm not sure. Do I do something with the -20 answer I got previously?

Answer 18

2.2 × 8.94= 19.66

Answer 19

Oh!!! I forgot something major. :(

Answer 20

It's the MAGNITUDE of the cross product! \[|\vec u \times \vec v| = |\vec u||\vec v|\sin(\theta)\]

Answer 21

And so you can test whether the angle between the perpendicular vectors is orthogonal by taking the magnitude of their cross product. \[|\vec u \times \vec v| = 0 \implies \vec u~ ||~\vec v\]

Answer 22

Now I'm confuse...

Answer 23

You've found \(|\vec u|\) and \(|\vec v|\), now find \[\vec u \cdot \vec v\]\[|\vec u \times\vec v|\]

Answer 24

19.66?

Answer 25

And what exactly is 19.66....?

Answer 26

What do you mean?

Answer 27

\(\color{blue}{\text{Originally Posted by}}\) @allydiaz
19.66?
\(\color{blue}{\text{End of Quote}}\)

What are you referring to?

Answer 28

Sorry your reply looks weird on my computer

Answer 29

refresh maybe?

Answer 30

Is there something I do next after 19.66. Or does that mean it's orthogonal ?

Answer 31

How did you get 19.66? It looks like a random number. What part did you solve to get 19.66? Thats what I am asking you to clarify.

Answer 32

u=(1,2), |u|= root(5)
v=(-4,8), |v| = root (80)

u . v = |u| |v| cos t

cos t = {u . v }/ {|u| |v|}

u . v = -4 + 16 = 12

cos t = 12 / { root(400) } = 3/5

t = ... doesn't look like they are parallel or ortogonal..

Answer 33

Yep, that's what I got as well, I was looking for the OP to solve the equation.

Answer 34

U × v= 2.23 × 8.94

Answer 35

The 2 is suppose to be a negative 2. I stated stated that i was typo in an earlier post

Answer 36

@Jhannybean and @IrishBoy123

Answer 37

So it doesn' t matter that its u=(1,-2) ?

Answer 38

"So in order to determine this, you need to use the dot product and the cross product of vectors that give you an angle measurement. "

pasting formulae off the internet is not going to help anyone.

@allydiaz
look at my last post. you only need to use the scalar/dot product.

Answer 39

OK

Answer 40

I still get my same answer : (

Answer 41

of course it matters. massively.

u = <1, -2>; |u| = root(5)
v = < -4, 8>, |v| = root ( 80)

u.v = -4 - 16 = -20 = root(400) cos t = 20 cost

cost = -1

t = - 180 deg. pointing in different directions.

these things are parallel!!

all you need to understand is the dot product to get that.

Answer 42

Thank you!! That's what I got as well

Answer 43

cool!