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11.4 The Cross product For an animation of this topic visit: / http://www.math.umn.edu/~nykamp/m2374/readings/crossprod Red x Green = Blue What is the cross product? • a × b is a vector that is perpendicular to both a and b. • ||a × b|| is the area of the parallelogram spanned by a and b (i.e. the parallelogram whose adjacent sides are the vectors a and b). • The direction of a×b is determined by the righthand rule. (This means that if we curl the fingers of the right hand from a to b, then the thumb points in the direction of a × b.) Definition of Cross Product of Two Vectors in Space Find the cross product of vectors (by hand and with a calculator) u = 3i +2j – k v = i + 2k Note: To find a cross product on the TI89 Press 2nd 5 (math) – 4 matrix – L Vector ops - crossP crossP([3,2,-1],[1,0,2]) Definition of Cross Product of Two Vectors in Space For an explanation and animation of the cross product visit: http://www.math.umn.edu/~nykamp/m2374/readings/ crossprod/ Example 1 • • • • • Find the cross product of the two vectors u = i – 2j + k v = 3i + j – 2k Find u x v Find v x u Find v x v Example 1 • • • • • Find the cross product of the two vectors u = i – 2j + k v = 3i + j – 2k Find u x v Find v x u Find v x v Example 1 continued • u = i – 2j + k v = 3i + j – 2k • Find a unit vector that is orthogonal to u and v The Triple Scalar Product Geometric Property of Triple Scalar Example 5 • • • • u = 3i – 5j + k v = 2j – 2k w = 3i + j + k Find the volume of the parallelepiped having vectors u, v and w for sides Example 5 hint This works because bxc yields a vector perpendicular to a and b in with the magnitude of the area of the parallelogram formed (the base of the parallel piped) bxc points in the direction of the height of the parallelepiped when dotted with a this gives the magnitude of bxc times the portion of a that points in the direction of bxc (the direction of the height )in other words this gives us the same as the formula Bh Q: Why should you never make a math teacher angry? A: You might get a cross product Q: What do you get when you cross an elephant and a banana? A: | elephant | * | banana | * sin(theta) Proof of the cross product proof that the cross product is orthogonal to the two original vectors is part of the homework │u x v │ = │u │ │v │sinθ