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Algebra for Athletes.com

8. Vectors

In the section on speed, we saw how lines can be used to relate time and distance.  Lines can also be used to show travel in two directions. 

In the speed section, we described speed by saying the receiver ran at 8 yards per second.  Speed only describes the magnitude of the motion. To say the receiver traveled at 8 yards per second straight downfield also describes the direction of the motion.  When direction is included with the speed it is called velocity. Velocity is called a vector quantity.

8.1 Displacement Vectors

A great way to describe vectors is with a golf example.  Irene, Jane, and Kate are playing a round of golf.  To describe their shots, weíre going to place a hole on an xy plane with the tee off area at the origin.  The cup is at a point 240 yards to the east and 180 yards to the north of the tee off area.  In other words, the cup has the coordinates of (240,180).  On the tee off, Irene hits the ball 223 yards to the east and 115 yards to the north.  Her drive could be shown by a vector, D, that looks like this. 

The vector can be described with numbers.  Instead of x and y, vectors use the letters i and j.  So the drive D is described as 223i + 115j.  This type of vector is known as a displacement vector because it shows how much the ball has been displaced during the shot.  On her next shot, the chip shot C, Irene hits the ball giving it a displacement vector of 33i+ 61j.  On her putt P, Irene sinks it with a vector of -16i + 4j.  If you add up the vectors, you get a vector of 240i + 115j.  This is the vector of a hit that would make a hole in one.  When you add several vectors together, you get a resultant vector, R. 

Of course, Jane and Kate have been playing too.  Their shots are shown in this figure.

Since Jane and Kate start at the same point and end up at the same point, their resultant vectors are the same as Irene's.

Since very few golf courses are perfectly flat, a more detailed description of the shots would include the distance up or down that each shot traveled.  Vectors are also used to describe motion in 3 dimensions.  In three dimensions, a k term is added. For example, if Irene's drive had landed on a slope 4 yards above the tee off area, the drive would be described with the vector D = 223i + 115j + 4k. 

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