Question

Are the vectors `u=<1, -2>` and `v=<4, 8>` parallel, orthogonal, or neither?

Answer 1

Neither parallel nor orthogonal. The angle between the two vectors is:

`theta ~~ 127^@` or `2.21`radians

Explanation:

Given: `vecu=<1, -2>` and `vecv=<4, 8>`

Compute the dot-product:

`vecu*vecv= (1)(4)+(-2)(8) = -12`

We computed the dot-product by multiplying the respective components of the vectors. Another definition of the dot-product is the magnitudes of the vectors multiplied by the angle between them:

`vecu*vecv= |vecu||vecv|cos(theta)" [1]"`

Compute the magnitudes of the two vectors:

`|vecu| = sqrt(1^2+(-2)^2) = sqrt(5)`
`|vecv|=sqrt(4^2+8^2) = sqrt(80)`

Substitute the values for the dot-product and the magnitudes into equation [1]:

`-12= sqrt(5)sqrt(80)cos(theta)`

`-12= sqrt(400)cos(theta)`

`-12= 20cos(theta)`

`cos(theta) = -12/20`

`theta = cos^-1(-3/5)`

`theta ~~ 127^@` or `2.21`radians