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Two vectors lying in the xy-plane are given by the equations A = 8i + 2j and B = -31 + 3j. Find Ax B and verify that A xB = -B x A. SOLUTION Conceptualize Given the unit-vector notations of the vectors, think about the directions the vectors point in space. Draw them on graph paper and imagine the parallelogram for these vectors. Categorize Because we use the definition of the cross product discussed in this section, we categorize this example as an analysis problem. Write the cross product of the two vectors: AXB = i+ 2j i+ 3j Perform the multiplication: AxB = 8i x (-31) + 8î x 3j + jx (-31) + 2j x Use the equations for the cross product of unit vectors to evaluate the various terms: ĀxB = To verify that AxB = -B x A, evaluate Bx A: BxA = (-31 + x (8i + Perform the multiplication: BxA = (-31) x li +(-3î) x 2j + 3j x 8i +| jx 2) Use the equations for the cross product of unit vectors evaluate the various terms: BxA = Therefore, A x B = -B xA As an alternative method for finding Ax B, you could use AXB - (AB - AB i+ y Z A_B_ - AB+ (A"y (A_8, - A,B ). zy z X Try it! EXERCISE What is the vector product ofA = 41 + 3j - sk and B = -6i + 8j + 7k? (Express your answer in vector form.) Hint AxB -

Two vectors lying in the xy-plane are given by the equations A = 8i + 2j and B = -31 + 3j. Find Ax B and verify that A xB = -B x A. SOLUTION Conceptualize Given the unit-vector notations of the vectors, think about the directions the vectors point in space. Draw them on graph paper and imagine the parallelogram for these vectors. Categorize Because we use the definition of the cross product discussed in this section, we categorize this example as an analysis problem. Write the cross product of the two vectors: AXB = i+ 2j i+ 3j Perform the multiplication: AxB = 8i x (-31) + 8î x 3j + jx (-31) + 2j x Use the equations for the cross product of unit vectors to evaluate the various terms: ĀxB = To verify that AxB = -B x A, evaluate Bx A: BxA = (-31 + x (8i + Perform the multiplication: BxA = (-31) x li +(-3î) x 2j + 3j x 8i +| jx 2) Use the equations for the cross product of unit vectors evaluate the various terms: BxA = Therefore, A x B = -B xA As an alternative method for finding Ax B, you could use AXB - (AB - AB i+ y Z A_B_ - AB+ (A"y (A_8, - A,B ). zy z X Try it! EXERCISE What is the vector product ofA = 41 + 3j - sk and B = -6i + 8j + 7k? (Express your answer in vector form.) Hint AxB -

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The Vector Product Two vectors lying in the xy-plane are given by the equations A = 8î + 2j and B = -3î + 3j. Find Āx Band verify that A x B = -B x A. SOLUTION Conceptualize Given the unit-vector notations of the vectors, think about the directions the vectors point in space. Draw them on graph paper and imagine the parallelogram for these vectors. Categorize Because we use the definition of the cross product discussed in this section, we categorize this example as an analysis problem. Write the cross product of the two vectors: AXB = Î + 2j) |i + 31 Perform the multiplication: AxB = 8i x (-31) + 8î x 3j + jx (-3î) + 2j > Use the equations for the cross product of unit vectors to evaluate the various terms: ĀXB = To verify that A × B = -B x A, evaluate Bx A: BxA = Perform the multiplication: BxA = (-31) x +(-31) x 2j + 3j × 8î + ]î x 2j Use the equations for the cross product of unit vectors evaluate the various terms: BxA = Therefore, Ax B = -B xA As an alternative method for finding Ax B, you could use AxB = AB - AB i + (AB - A B j+ y z Try it! EXERCISE What is the vector product of A = 4î + 3j - 5k and B = -6î + 8j + 7k? (Express your answer in vector form.) Hint AxB = Need Help? Read It
Transcribed Image Text:The Vector Product Two vectors lying in the xy-plane are given by the equations A = 8î + 2j and B = -3î + 3j. Find Āx Band verify that A x B = -B x A. SOLUTION Conceptualize Given the unit-vector notations of the vectors, think about the directions the vectors point in space. Draw them on graph paper and imagine the parallelogram for these vectors. Categorize Because we use the definition of the cross product discussed in this section, we categorize this example as an analysis problem. Write the cross product of the two vectors: AXB = Î + 2j) |i + 31 Perform the multiplication: AxB = 8i x (-31) + 8î x 3j + jx (-3î) + 2j > Use the equations for the cross product of unit vectors to evaluate the various terms: ĀXB = To verify that A × B = -B x A, evaluate Bx A: BxA = Perform the multiplication: BxA = (-31) x +(-31) x 2j + 3j × 8î + ]î x 2j Use the equations for the cross product of unit vectors evaluate the various terms: BxA = Therefore, Ax B = -B xA As an alternative method for finding Ax B, you could use AxB = AB - AB i + (AB - A B j+ y z Try it! EXERCISE What is the vector product of A = 4î + 3j - 5k and B = -6î + 8j + 7k? (Express your answer in vector form.) Hint AxB = Need Help? Read It
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