The Physics
Hypertextbook
Opus in profectus

# Vector Resolution and Components

## Summary

• A vector (r) in a plane can be represented in one of two equivalent ways:
• in polar coordinates as an overall magnitude with a direction relative to an agreed upon direction (typically identified as the x-axis).

r = r  + θ θ̂

where…  r = magnitude θ = direction angle r̂ = unit vector in the radial direction θ̂ = unit vector in the angular direction
• in rectangular coordinates (cartesian coordinates) as a pair of components, one for each axis

r = x  + y

where…  x = component in the x-direction y = component in the y-direction î = unit vector in the x-direction ĵ = unit vector in the y-direction
• Converting between representations
• A vector with a known magnitude and direction can be resolved into its x and y components using the definitions of cosine and sine, respectively. This process is called resolution.

x = r cos θ

y = r sin θ

• The components of a vector can be added together find the vector's magnitude and direction using pythagorean theorem and tangent, respectively. This is just vector addition for two perpendicular vectors.
 r2 = x2 + y2
 tan θ = y x
• Uniqueness of values
• The values of x, y, and θ depend on the orientation of the coordinate axes chosen. They are relative quantities.
• The value of r does not depend on the orientation of the coordinate axes chosen. It is an invariant quantity.