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 r̂ + 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 î + y ĵ
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
- in polar coordinates as an overall magnitude with a direction relative to an agreed upon direction (typically identified as the x-axis).
- 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
- 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.
- 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.