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chap12sec4

# chap12sec4 - P(1,0-1,Q(2,4,5 and R(3,1,7 and find the area...

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12.4 The Cross Product The cross product is a  vector,  sometimes called the  vector product.  Cross products are only found with 3-dimensional  vectors. Cross Product 1, 2 3 1 2 3 2 3 3 2 3 1 1 3 1 2 2 1 , , , , , a a a b b b a b a b a b a b a b a b = = × = - - - a b a b Determinants can be used to find this product. The TI-85 and TI-86 will also perform this operation. Determinants and Cross Product     1 2 3 1 2 3 2 3 1 3 1 2 2 3 1 3 1 2 1 2 3 1 2 3 , , , , a b ad bc c d a a a b b b a a a a a a b b b b b b a a a b b b = - = = × = - + × = a b a b i j k i j k a b Example: Find the cross product for  1,2, 3 5, 1, 2 = - = - - a b . Theorems, etc.    (a)  The cross product is orthogonal to both  a  and  b. (b)  If  θ  is the angle between  a  and  b then  sin θ × = a b a b . (c)  Two nonzero vectors  a  and  b are parallel if and only if the cross product is 0.

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(d) The length of the cross product is equal to the area of the parallelogram determined        by  a  and  b . Example: Find a vector orthogonal to the plane through the points
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Unformatted text preview: P(1,0,-1) ,Q(2,4,5), and R(3,1,7) and find the area of the triangle formed by the points. Look at the properties in Theorem 8 on page 807. Property (5) is called the scalar triple product of the vectors a , b and c and can be written as a determinant. This product is used to find the volume of a parallelepiped. ( 29 1 2 3 1 2 3 1 2 3 a a a b b b c c c ⋅ × = a b c The volume of the parallelepiped determined by the vectors a , b and c is the magnitude of their scalar triple product. a (b c) V = ⋅ × Example: Find the volume of the parallelepiped determined by the points P(0,1,2), Q(2,4,5), R(-1,0,1) and S(6,-1,4). a b c a x b θ...
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chap12sec4 - P(1,0-1,Q(2,4,5 and R(3,1,7 and find the area...

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