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Vectors are used to denote or model directions.
Vectors can be added graphically: draw one vector on paper, move the other so that its tail coincides with the head of the first. Vectors can also be added by components. Just add the corresponding components together. For example, if one vector is (10, 0) and the other is (0, 5) (those two would be perpendicular), the combined vector is (10+ 0, 0 + 5), that is, (10, 5). Such a vector can also be converted to polar coordinates, that is, a length and an angle; use the "rectangular to polar" conversion on your scientific calculator to do that.
Vectors have a lot of applications in physics and engineering. Velocity, acceleration and forces are vectors. For example, you can use vectors to determine the total distance travelled by a boat travelling across a river at 25 miles/hour while the river flows perpendicular to the boat at 10 miles/hour. There are loads of other applications. Vectors can be used to predict the movement of planets in their orbits, and are useful for 3D geometry in general. Vectors are also essential in electromagnetics, where they are used to represent electric and magnetic fields and their relationships. Maxwell's equations are equations which use vectors. The Poynting vector is used to calculate the flow of electromagnetic energy through space, for example by radio waves or light.
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You must find the x and y components of each vector. Then you add up the like x components and the like y components. Using your total x component and total y component you may then apply the pythagorean theorem.