Vectors - Vectors There is a great temptation to put...

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There is a great temptation to put vectors tail-to-tail when you go to add them. May all the battles that you wage in your war against that temptation end with your glorious triumph. Vectors add head-to-tail. We are about to embark on the study of the motion of a particle which does not move only along a line. To be sure, we will take things a step at a time and next study the motion of a particle that stays in a plane, an imaginary flat (but not necessarily horizontal) sheet. In considering such motion we find that we can no longer specify the direction of a quantity that has direction by means of a simple + or – sign as we could in the case of motion along a line. Now, the direction of a quantity that has direction, such as the velocity of a car moving along a horizontal surface, might be specified in words as, for example, “in a compass direction that is 18 degrees north of due east”. A number, with or without a plus or minus sign, is no longer sufficient to specify the magnitude and direction of a velocity, or an acceleration. Enter the vector. The vector is a mathematical entity 1 that has both magnitude and direction. Mathematicians have devised some operations involving vectors that pertain to physical quantities that have magnitude and direction. Knowing about vectors, and the mathematical operations devised for them, comes in mighty handy in the study of physics. We start our discussion of vectors by presenting a couple of different representations of vectors (graphical and magnitude-and-direction). Graphical Representation of a Vector Graphically, a vector is represented by an arrow. The magnitude of the vector is represented by the length of the arrow and the direction of the vector is represented by which way the arrow is pointing. A vector variable is represented by a letter with an arrow over it 2 as in the vector A . Once we define a vector, we often need to write about the magnitude of that vector—just how big it is as opposed to both how big and which way it is. The magnitude of the vector A can be written two ways, either A or A . The first way makes it more obvious that we are dealing with the magnitude of a vector (rather than some ordinary variable). The second method is used when it is already clear from the context that we are dealing with the magnitude of a vector. You can never go wrong using A . A is easier to write but the reader might think that it is an ordinary variable rather than the magnitude of a vector. 1 An entity is just a “something”. Thus, when we say that a vector is a mathematical entity, all we are saying is that a vector is a mathematical “something”. I use the word entity here because one typically uses “a something” when one is talking about a solid object, and “an entity” when one is talking about a “non-object” such a thought or a ghost or, as in the case at hand, a vector. 2
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Vectors - Vectors There is a great temptation to put...

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