Using Trigonometry to Break Vectors into Components



Using Trigonometry to Break Vectors into Components

When learning the basics about vectors and how they work, it’s important to understand how to manipulate them into forms you can use.

The Basics

Vectors have a magnitude and phase, as mentioned in a previous tutorial. The magnitude is essentially the length of the vector and the phase gives the direction (aka. the angle at which the vector is drawn). We can also think of vectors in their component form (x-component and y-component). So how do we get back and forth from these two different forms?

Magnitude-Phase Form to Component Form

The x and y components are related to magnitude and phase via the following equations.

x-component = magnitude * cos(phase)

y-component = magnitude * sin(phase)

Example 1: Converting from Magnitude and Phase Form to Component Form

A vector has a magnitude of 4 and a phase of 35 degrees. What are its x and y-components?

Using the above formulas we have

x = 4 * cos(35) = 3.2766

y = 4 * sin(35) = 2.2943

Component Form is also sometimes referred to as “Rectangular” or “Cartesian” Form.

Component Form to Magnitude-Phase Form

As before there are two equations to relate these quantities

Magnitude = [pic]

Phase = [pic]

Where x is the x-component and y is the y-component. Also tan-1 means arctan.

Example 2: Converting from Component Form to Magnitude and Phase Form

A vector has an x-component of 4 units and a y component of 6 units. What are its magnitude and phase?

Magnitude = [pic]

Phase = [pic]degrees.

Magnitude and Phase Form is also known as Polar Form.

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