Explanation Given information: Consider that the given vectors . v = i + j + 2 k (1) u = − i − k (2) Calculation: Calculate the magnitude of vector v using Equation (1). | v | = | i + j + 2 k | = 1 2 + 1 2 + 2 2 = 6 Calculate the magnitude of vector u using Equation (2). | u | = | − i − k | = ( − 1 ) 2 + ( − 1 ) 2 = 2 Calculate the value of v ⋅ u . v ⋅ u = ( i + j + 2 k ) ⋅ ( − i − k ) = − 1 + 0 − 2 = − 3 Calculate the value of u ⋅ v . u ⋅ v = ( − i − k ) ⋅ ( i + j + 2 k ) = − 1 + 0 − 2 = − 3 Calculate the cross product v × u using Equations (1) and (2). v × u = ( i + j + 2 k ) × ( − i − k ) = | i j k 1 1 2 − 1 0 − 1 | = i ( − 1 − 0 ) − j ( − 1 + 2 ) + k ( 0 + 1 ) v × u = − i − j + k (3) Consider the expression for the cross product u × v . u × v = − ( v × u ) (4) Substitute Equation (3) in (4). u × v = − ( − i − j + k ) = i + j − k Calculate the magnitude of Equation (3). | v × u | = | − i − j + k | = ( − 1 ) 2 + ( − 1 ) 2 + ( 1 ) 2 = 3 Write the expression for acute angle between the vectors. θ = cos − 1 ( v ⋅ u | v | | u | ) (5) Substitute Equations (1) and (2) in (5)

University Calculus: Early Transcendentals, Loose-leaf Edition (4th Edition)

4th Edition

ISBN: 9780135164860

Author: Joel R. Hass, Christopher Heil, Przemyslaw Bogacki, Maurice D. Weir, George B. Thomas Jr.

Publisher: PEARSON

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Chapter 11, Problem 18PE

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Calculate the value of |v|, |u|, v⋅u, u⋅v, v×u, u×v, |v×u|, the angle between v and u, the scalar component of u in the direction of v, and the vector projection of u onto v.

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Chapter 11, Problem 17PE

Chapter 11, Problem 19PE

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Chapter 11 Solutions

University Calculus: Early Transcendentals, Loose-leaf Edition (4th Edition)

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Calculate the value of | v | , | u | , v ⋅ u , u ⋅ v , v × u , u × v , | v × u | , the angle between v and u , the scalar component of u in the direction of v , and the vector projection of u onto v .

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Calculate the value of |v|, |u|, v⋅u, u⋅v, v×u, u×v, |v×u|, the angle between v and u, the scalar component of u in the direction of v, and the vector projection of u onto v.

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Chapter 11 Solutions