Question

How do I find the angle between vectors `<3, 0>` and `<5, 5>`?

Answer 1

Let's begin by naming our vectors.

v`=<3,0>`
w`=<5,5>`

Dot product `-> v*w`

Angle between formula `->cos (theta)=(v*w)/(||v||*||w||)`

Solve for `theta`

`cos^-1(cos (theta))=cos^-1((v*w)/(||v||*||w||))`

`theta=cos^-1((v*w)/(||v||*||w||))`

Begin by finding the dot product of vectors `v and w` by adding the products of the horizontal and vertical components.

`theta=cos^-1(((3)(5)+(0)(5))/(||v||*||w||))`

`theta=cos^-1((15+0)/(||v||*||w||))`

Now find the magnitudes of both vectors

`theta=cos^-1((15)/(sqrt(9+0)*sqrt(25+25)))`

`theta=cos^-1((15)/(sqrt(9)*sqrt(50)))`

`theta=cos^-1((15)/(sqrt(3*3*5*5*2)))`

`theta=cos^-1((15)/(15sqrt(2)))`

`theta=cos^-1((1)/(sqrt(2)))`

`theta=pi/4`