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Unformatted text preview: Harvey Mudd College Math Tutorial: Elementary Vector Analysis In order to measure many physical quantities, such as force or velocity, we need to determine both a magnitude and a direction. Such quantities are conveniently represented as vectors. The direction of a vector v in 3-space is specified by its components in the x , y , and z directions, respectively: ( x,y,z ) or x i + y j + z k , where i , j , and k are the coordinate vectors along the x , y , and z-axes. i = (1 , , 0) j = (0 , 1 , 0) k = (0 , , 1) The magnitude of a vector v = ( x,y,z ), also called its length or norm , is given by k v k = q x 2 + y 2 + z 2 . Notes Vectors can be defined in any number of dimensions, though we focus here only on 3-space. When drawing a vector in 3-space, where you position the vector is unimportant; the vectors essential properties are just its magnitude and its direction. Two vectors are equal if and only if corresponding components are equal. A vector of norm 1 is called a unit vector . The coordinate vectors are examples of unit vectors. The zero vector, = (0 , , 0), is the only vector with magnitude 0. Basic Operations on Vectors To add or subtract vectors u = (...
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- Summer '11