so far - So far in our course Important theoretical...

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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 non-zero vector, then v / || v || is a unit vector. All the above concepts generalize trivially to 3-dimensional vectors. The dot product of two 3-dimensional 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 non-zero vector v is given by the formula: proj v ( u ) = u v v v v The component of a vector u along a non-zero 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 right-handed 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 non-zero 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.

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so far - So far in our course Important theoretical...

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