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Unformatted text preview: So far in our course Important theoretical concepts The vector v that has originates at the point ( a 1 ,b 1 ) and terminates at the point ( a 2 ,b 2 ) has components < a 2 a 1 ,b 2 b 1 > . The magnitude of the vector v = < a,b > equals:  v  = p a 2 + b 2 The sum of the vectors u = < u 1 ,u 2 > and v = < v 1 ,v 2 > has compo nents: u + v = < u 1 + v 1 ,u 2 + v 2 > The scalar multiple v of the vector v = < v 1 ,v 2 > has components: v = < v 1 ,v 2 > The vectors u and v are parallel if and only if there exists real number such that u = v or v = u . A vector u is unit if  u  = 1. If v is a nonzero vector, then v /  v  is a unit vector. All the above concepts generalize trivially to 3dimensional vectors. The dot product of two 3dimensional vectors u = < u 1 ,u 2 ,u 3 > and v = < v 1 ,v 2 ,v 3 > is defined by the formula: u v = u 1 v 1 + u 2 v 2 + u 3 v 3 The dot product is linear with respect to each of the two variables and commutative. Also, notice that uu =  u  2 for any vector u . Two vectors are orthogonal to each other if and only if their dot prod uct is 0. 1 The angle between two vectors u and v is defined by the formula: cos = u v  u  v  The projection of a vector u onto a nonzero vector v is given by the formula: proj v ( u ) = u v v v v The component of a vector u along a nonzero vector v is given by the expression: u v  v  For any two vectors u and v we have the following inequality:  u v   u  v  The equality holds true if and only if u and v are parallel. The cross product of two three dimensional vectors u = < u 1 ,u 2 ,u 3 > and v = < v 1 ,v 2 ,v 3 > is defined by the formula: u v = i j k u 1 u 2 u 3 v 1 v 2 v 3 The vector u v is perpendicular to both u and v . Also, the vectors u , v , u v form a righthanded system. The cross product is linear with respect to each of the two variables but it is not commutative. In fact, it satisfies the relation: u v = ( v u ) Given 3 vectors u = < u 1 ,u 2 ,u 3 > , v = < v 1 ,v 2 ,v 3 > and w = < w 1 ,w 2 ,w 3 > , their mixed product is defined by the formula: ( u v ) w 2 It is actually a number and it equals the determinant: u v = u 1 u 2 u 3 v 1 v 2 v 3 w 1 w 2 w 3 The absolute value of this determinant equals the volume of the par allelepiped formed by the three vectors. The magnitude of the cross product of the vectors u and v equals the area of the parallelogram formed by these two vectors. In particular:  u v  =  u  v  sin where is the angle between the two vectors. The equation of the plane that goes through the point ( x o ,y o ,z o ) and is perpendicular to the nonzero vector < a,b,c > is given by: < a,b,c. > < x x o ,y y o ,z z o > = 0 or equivalently: a ( x x o ) + b ( y y o ) + c ( z z o ) = 0 If we wish to find the equation of a plane that goes through three points, then all we need to do is find two linearly independent vectors...
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This note was uploaded on 10/27/2009 for the course MATH 310 taught by Professor Smith during the Spring '08 term at Ill. Chicago.
 Spring '08
 Smith

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