chap12sec4

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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
(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 
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This note was uploaded on 01/20/2012 for the course MATH 2057 taught by Professor Estrada during the Fall '08 term at LSU.

Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online