Cross product of two vectors
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Cross product of two vectors
In two dimensions any two vectors define a plane. [Diagram goes here - download the original to see it.] We wish to find a vector that is perpendicular to the two vectors a and b. This vector is the cross product of a and b and is denoted a ´ b [Diagram goes here - download the original to see it.] Let be the angle between the vectors a and b. Note that need not be a right angled triangle. Then a x b is defined to be [Equation goes here - download the original to see it.] where [Equation goes here - download the original to see it.] is a unit vector perpendicular to both a and b. We find the cross product of two vectors in three dimensions by a technique of expanding a determinant as follows. [Equation goes here - download the original to see it.] Then [Equation goes here - download the original to see it. Example [Equation goes here - download the original to see it.] Solution [Equation goes here - download the original to see it.] To confirm that this is the case we will find the dot (scalar) products of a ´ b with a and b separately: [Equation goes here - download the original to see it.] Similarly [Equation goes here - download the original to see it.] When the cross product has not been studied a perpendicular vector to two vectors must be found by using the properties of the dot product and solving a system of simultaneous equations. Example For example, find a vector perpendicular to a and b when [Equation goes here - download the original to see it.] Solution [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.]Hence [Equation goes here - download the original to see it.] is perpendicular to both a and b. The technique using determinants is easier if you know it! The perpendicular vector to a and b is not unique. Any scalar multiple of a vector perpendicular to a and b is also perpendicular to a and b. [Diagram goes here - download the original to see it.]
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Contents of
Cross product of two vectors
1 Cross product of two vectors
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