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Learning Introductory Physics with Activities

Paul J. Emigh, Rebecka Tumblin, Kathryn Hadley, Danielle Skinner

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Section 1.5 Vector Components

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One useful property of vectors is that they can be broken down into components along the axes of a coordinate system using unit vectors.
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Consider a position vector r→ shown in the two-dimensional Cartesian coordinate system below.
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Figure 1.5.1. A representation of a position vector in an xy-coordinate system showing the positive angle from the x-axis and the components of r→ along the x- and y-axes .
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You can write r→ in terms of its components in the x- and y-directions using unit vectors.
r→=rxx^+ryy^
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The scalar quantities rx and ry are called the components of r→. When multiplied by the unit vectors x^ and y^ they form a vector quantity. The scalar quantities rx and ry tell us how far we must move in the x- and y-direction to be at position r→.
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The magnitude, r, of vector r→ (magnitudes will be written without the vector arrows overhead) is the hypotenuse of a right triangle with legs of length rx and ry.
r=rx2+ry2
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Magnitudes are also sometimes written symbolically using bars |r→| around the vector. Both representations are equivalent.
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Exercises Practice Activities

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1.

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Sketch four different vectors so that each vector points in a different quadrant 1 . For each vector, choose an angle and sketch a right triangle whose legs are the components of the vector, like the one in the figure above.
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References References

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[1]
  
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Vector Decomposition into components by Dr. Michelle Tomasik from MIT 8.01 Classical Mechanics, Fall 2016, used under Creative Commons BY-NC-SA.
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[2]
  
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Magnitude and Direction by Dr. Michelle Tomasik from MIT 8.01 Classical Mechanics, Fall 2016, used under Creative Commons BY-NC-SA.
the vector shown above is in the first quadrant; the second quadrant has a negative x-component; the third quadrant has both components negative; and the fourth quadrant has negative y-component
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