A basic operation is to split a vector into components with respect to a coordinate system. The x-component gives the amount of the vector in the x-direction and the y-component gives the amount in the y-direction.

Take: +x = right and +y = up. To find the components v, draw this vector as the hypotenuse of a right triangle, with the other two sides parallel to the x and y axes. Then:

• Use geometry to find the magnitude of each component
• Use a diagram to determine the sign of each component

Horizontal component:

cos(θ)

=

vx v

, so

vx = v cos(θ)

Including the positive direction from the diagram, v_x = +v cos(θ)

Vertical component:

sin(θ)

=

vy v

, so

vy = v sin(θ)

Including the negative direction from the diagram, v_y = -v sin(θ)

If v = 3.5 and θ = 25 degrees, then:   v_x = 3.5 cos(25) = 3.2       v_y = -3.5 sin(25) = -1.5

Final result:   v = 3.2 -1.5