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Short Course:  Introduction to Vectors - Part I

Breaking a vector into x and y components


Often we would like to be able to convert from polar coordinates to cartesian coordinates when we represent a vector.  Using some simple trig we can do this.  Here we take you step-by-step through the procedure.

   
Labeling the sides of a vector

Step 1 - Label sides

The first step is to label the sides of the right-triangle that our vector creates with the x-axis.  Remember from basic trigonometry that the side opposite over the hypotenuse is called the sine of the angle.  The side adjacent over the hypotenuse is called the cosine of the angle.  Once we know this it's easy to convert from polar coordinates to cartesian coordinates.

Definition of the trigonometric function sine
Definition of the cosine of an angle

Step 2 - Remember some basic Trig

Recall from basic trig that the side opposite over the hypotenuse is called the sine of the angle.  Similarly, the side adjacent over hypotenuse is called the cosine of the angle.  We can use these formulas to calculate the x and y components of the vector.

Find the x-component (projection) of a vector

Step 3 - Calculate x

Looking at the formulas for sine and cosine it should be apparent that the x-component of the vector is simply the length of the side adjacent.  The length of the vector is just the length of the hypotenuse.  Using this knowledge we can calculate x.

Length of a two-dimensional vector along the y-axis

Step 4 - Calculate y

The y-component formula is derived in a similar fashion as shown.



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