Resolving vectors into components

Any vector A in a plane can be written as λa + μb along two chosen non-parallel vectors a and b; λa and μb are its components along them.

Unit vectors î, ĵ, k̂ have magnitude 1, point along +x, +y and +z, and have no dimension or unit. Any vector A equals |A| n̂, where n̂ is the unit vector along A.

A vector of magnitude A at angle θ to the x-axis has components Ax = A cos θ and Ay = A sin θ, so A = Ax î + Ay ĵ. A component can be positive, negative or zero.

From the components back to the vector: A = √(Ax² + Ay²) and tan θ = Ay/Ax. A ball thrown at 20 m/s at 30° has components 17.3 m/s and 10.0 m/s.

In three dimensions, Ax = A cos α, Ay = A cos β, Az = A cos γ, where α, β, γ are the angles with the axes, and A = √(Ax² + Ay² + Az²).

The position vector of a point (x, y, z) is r = x î + y ĵ + z k̂.