by Jeffrey Graf

Created on: February 27, 2010   Last Updated: March 02, 2010

A vector is a mathematical object that can be used to represent a physical or mathematical property. By object, I mean a construct that contains two or more components. A vector always has a magnitude and a direction. Vectors are useful for representing forces,  velocities, orientations, phases, and very often rays of light.

Operations on a vector.

The first example I will use is addition of two vectors. Suppose we have a rail car and two locomotives, each capable of pulling with a force of L. If we position the two locomotives in tandem to each other, they will be capable of pulling our rail car with a force of 2(L) . Suppose we position our locomotives so that one is pulling in the forward direction and one is pulling in the reverse direction. Obviously, they will cancel each other out, and the resulting force will be zero. To represent this mathematically, we let the track represent the x axis. When our two locomotives are pulling in the same direction we sum their magnitudes, or L, but when one is reversed, we represent its force generation by negative L, so that when summed, it gives a resulting value of 0.

This one dimensional example is easily extendible to two dimensions. Suppose now instead of a train on a track, we are in a boat on a sea. Now we have an x axis which runs from bow to stern and a y axis that runs from starboard to port. Our presumed boat is a sailboat, which means a force is being exerted on it which has components in both directions, x and y. Likewise, the centerboard of the boat is exerting a force counter to the direction of  the wind, that will counter the boats tendency to drift downwind. As the boat moves forward in the water, the resistance due to the viscosity of the water increases until the boat reaches a maximum speed. When the boat is moving at a maximum speed, we can presume that the x components of all of the force vectors equals 0 and the y components of all of the force vectors equals 0. By creating this type of mathematical model for a physical system, like a sailing boat, we can make predictions about how changes in the size of the sail, the length of the boat or the size of the centerboard will affect overall speed. In fact, these types of predictions have given rise to sailboat classes, for example the 12-meter rule yachts. A 12-meter rule yacht is not 12 meters long ( about 30 feet ) but closer to 70 feet long.  The parameters of each design are plugged into an equation, and must produce a number

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