Unit Vectors and Vector Components

[Alexrey]

I remember in my first year physics classes, when dealing with a force F we would find the vector's x- and y-components using $$F_x=r cos(θ)$$ and $$F_y=r sin(θ)$$ I also remember learning in my mathematics classes about unit vectors, but cannot seem to remember using them to break vectors down into their respective x- and y-components. As such, I thought I'd ask this question: Am I correct in thinking that when using a Cartesian coordinate system to describe a force F, if we find the unit vector for F, are the x- and y- components of this unit vector equivalent to $$cos(θ)$$ and $$sin(θ)$$ respectively?

 

[Mark44]

I remember in my first year physics classes, when dealing with a force F we would find the vector's x- and y-components using $$F_x=r cos(θ)$$ and $$F_y=r sin(θ)$$ I also remember learning in my mathematics classes about unit vectors, but cannot seem to remember using them to break vectors down into their respective x- and y-components. As such, I thought I'd ask this question: Am I correct in thinking that when using a Cartesian coordinate system to describe a force F, if we find the unit vector for F, are the x- and y- components of this unit vector equivalent to $$cos(θ)$$ and $$sin(θ)$$ respectively?

Yes, assuming θ is the angle that F makes with the x-axis.

Think about it.
F_x = |F| cos(θ)
F_y = |F| sin(θ)

To get unit vectors, divide each of the above by its magnitude, |F|.

 

[Alexrey]

Great, thank you.