# cal2 - Chapter Two Vectors-Algebra and Geometry 2.1 Vectors...

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2.1 Chapter Two Vectors-Algebra and Geometry 2.1 Vectors A directed line segment in space is a line segment together with a direction. Thus the directed line segment from the point P to the point Q is different from the directed line segment from Q to P . We frequently denote the direction of a segment by drawing an arrow head on it pointing in its direction and thus think of a directed segment as a spear. We say that two segments have the same direction if they are parallel and their directions are the same: Here the segments L1 and L2 have the same direction. We define two directed segments L and M to be equivalent ( L M 2245 ) if they have the same direction and have the same length. An equivalence class containing a segment L is the set of all directed segments equivalent with L . Convince yourself every segment in an equivalence class is equivalent with every other segment in that class, and two different equivalence classes must be disjoint. These equivalence classes of directed line segments are called vectors . The members of a vector v are called representatives of v . Given a directed segment u , the vector which contains u is called the vector determined by u. The length , or magnitude , of a vector v is defined to be the common length of the representatives of v . It is generally designated by | v |. The angle between two vectors u and v is simply the angle between the directions of representatives of u and v .

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2.2 Vectors are just the right mathematical objects to describe certain concepts in physics. Velocity provides a ready example. Saying the car is traveling 50 miles/hour doesn’t tell the whole story; you must specify in what direction the car is moving. Thus velocity is a vector-it has both magnitude and direction. Such physical concepts abound: force, displacement, acceleration, etc. The real numbers (or sometimes, the complex numbers) are frequently called scalars in order to distinguish them from vectors. We now introduce an arithmetic, or algebra, of vectors. First, we define what we mean by the sum of two vectors u and v . Choose a spear u from u and a spear v from v . Place the tail of v at the nose of u . The vector which contains the directed segment from the tail of u to the nose of v is defined to be u v + , the sum of u and v . An easy consequence of elementary geometry is the fact that | u + v | < | u | + | v |. Look at the picture and convince yourself that the it does not matter which u spear or v spear you choose, and that u v v u + = + : Convince yourself also that addition is associative: u + ( v + w ) = ( u + v ) + w . Since it does not matter where the parentheses occur, it is traditional to omit them and write simply u + v + w . Subtraction is defined as the inverse operation of addition. Thus the difference
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## This note was uploaded on 10/27/2009 for the course MBA 1918272 taught by Professor Peter during the Fall '09 term at Aberystwyth University.

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cal2 - Chapter Two Vectors-Algebra and Geometry 2.1 Vectors...

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