Question

How do you determine whether u and v are orthogonal, parallel or neither given `u=<3, 15>` and `v=<-1, 5>`?

Answer 1

Please see the explanation.

Explanation:

Compute the dot-product:

`baru*barv = 3(-1) + 15(5) = 72`

The two vectors are not orthogonal; we know this, because orthogonal vectors have a dot-product that is equal to zero.

Determine whether the two vectors are parallel by finding the angle between them.

Compute the magnitude of both vectors:

`||baru|| = sqrt(3^2 + 15^2) = sqrt(234)`

`||barv|| = sqrt((-1)^2 + 5^2) = sqrt(26)`

The angle between them is:

`theta = cos^-1(72/(sqrt(234)sqrt(26)))`

`theta ~~ 22.6^@`

If they were parallel the angle would be `0^@ or 180^@`, therefore, the two vectors are not parallel.

The answer is neither.

Answer 2

The vectors are not parallel and not orthogonal.

Explanation:

To see if 2 vectors, we do a dot product

`vecu.vecv=〈3,15〉.〈-1,5〉=3*-1+15*5=-3+75=72`

As the dot product is `!=0`, the vectors are not perpendicular

If 2 vectors are parallel,

then, `vecu=kvecv`

`〈3,15〉=k〈-1,5〉`

`3=-k` and `15=5k`

Therefore, `k=-3` and `k=3`

This is not possible, so the vectors are not parallel