Find the component form of the sum of u and v with direction
angles θ_{u} and θ_{v}.

||u||=2

||v||=4

Θ_{u}=30°

Θ_{v}=135°

tan θ_{u}=tan 30°=1/√3=(1/2)/(√3/2)

Assume for now that ½=b and √3/2=a (<√3/2, ½>)… also is a unit vector because these are the coordinates associated with θ=30° on the unit circle.

||<√3/2, ½>||=1

For a magnitude ||u||=2,

2<√3/2, ½>=<√3, 1>

tan θ_{v}=tan
135°=-1=(√2/2)/(-√2/2)

Assume for now that √2/2=b and -√2/2=a (<-√2/2, √2/2>)… also is a unit vector because these are the coordinates associated with θ=135° on the unit circle.

||<-√2/2, √2/2>||=1

For a magnitude ||u||=4,

4<-√2/2, √2/2>=<-2√2, 2√2>

u+v=<√3, 1>+<-2√2, 2√2>=<√3-2√2, 1+2√2>