SECTION 9.7 The Cross Product 669 1 Find the Cross Product of Two Vectors For vectors in space, and only for vectors in space, a second product of two vectors is defined, called the cross product. The cross product of two vectors in space is also a vector that has applications in both geometry and physics. We first define the cross product algebraically. Later we will discuss some geometric properties. 9.7 The Cross Product OBJECTIVES 1 Find the Cross Product of Two Vectors (p. 669) 2 Know Algebraic Properties of the Cross Product (p. 670) 3 Know Geometric Properties of the Cross Product (p. 671) 4 Find a Vector Orthogonal to Two Given Vectors (p. 672) 5 Find the Area of a Parallelogram (p. 672) DEFINITION Cross Product If a b c v i j k 1 1 1 = + + and a b c w i j k 2 2 2 = + + are two vectors in space, the cross product v w× is defined as the vector b c b c a c a c a b a b v w i j k 1 2 2 1 1 2 2 1 1 2 2 1 ( ) ( ) ( ) × = − − − + − (1) Notice that the cross product v w× of two vectors is a vector. Because of this, it is sometimes referred to as the vector product . Finding a Cross Product Using Equation (1) If v i j k 2 3 5 = + + and w i j k 2 3 , = + + find v w. × Solution EXAMPLE 1 ( ) ( ) ( ) ( ) ( ) ( ) × = ⋅ − ⋅ − ⋅ − ⋅ + ⋅ − ⋅ = − − − + − = − − + v w i j k i j k i j k 3 3 2 5 2 3 1 5 2 2 1 3 9 10 6 5 4 3 Equation (1) Determinants* may be used as an aid in computing cross products. A 2 by 2 determinant , symbolized by a b a b 1 1 2 2 has the value a b a b ; 1 2 2 1 − that is, a b a b a b a b 1 1 2 2 1 2 2 1 = − A 3 by 3 determinant has the value = − + A B C a b c a b c b c b c A a c a c B a b a b C 1 1 1 2 2 2 1 1 2 2 1 1 2 2 1 1 2 2 *Determinants are discussed in detail in Section 11.3.
RkJQdWJsaXNoZXIy NjM5ODQ=