(a) Explanation Given information: The vectors are, u = i + 2 j − k v = − i + j + k w = i + k r = − ( π 2 ) i − π j + ( π 2 ) k Calculation: Finding the dot product of u ⋅ v . u ⋅ v = [ ( 1 ) ( − 1 ) ] + [ ( 2 ) ( 1 ) ] + [ ( − 1 ) ( 1 ) ] = − 1 + 2 − 1 u ⋅ v = 0 The dot product of u ⋅ v is equal to zero. Hence, they are perpendicular. Finding the dot product of u ⋅ w . u ⋅ w = [ ( 1 ) ( 1 ) ] + [ ( 2 ) ( 0 ) ] + [ ( − 1 ) ( 1 ) ] = 1 + 0 − 1 u ⋅ w = 0 The dot product of u ⋅ w is equal to zero. Hence, they are perpendicular. Finding the dot product of u ⋅ r . u ⋅ r = [ ( 1 ) ( − π 2 ) ] + [ ( 2 ) ( − π ) ] + [ ( − 1 ) ( π 2 ) ] = − π 2 − 2 π − π 2 = − π − 4 π − π 2 = − 6 π 2 u ⋅ r = − 3 π The dot product of u ⋅ r is not equal to zero (b) To determine Find whether the given vectors u , v , w , and r are parallel.

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 11.4, Problem 24E

(a)

To determine

Find whether the given vectors u, v, and w are perpendicular.

(b)

To determine

Find whether the given vectors u, v, w, and r are parallel.

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Chapter 11.4, Problem 23E

Chapter 11.4, Problem 25E

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u, v and w are three coplanar vectors: ⚫ w has a magnitude of 10 and points along the positive x-axis ⚫ v has a magnitude of 3 and makes an angle of 58 degrees to the positive x- axis ⚫ u has a magnitude of 5 and makes an angle of 119 degrees to the positive x- axis ⚫ vector v is located in between u and w a) Draw a diagram of the three vectors placed tail-to-tail at the origin of an x-y plane. b) If possible, find w × (ū+v) Support your answer mathematically or a with a written explanation. c) If possible, find v. (ū⋅w) Support your answer mathematically or a with a written explanation. d) If possible, find u. (vxw) Support your answer mathematically or a with a written explanation. Note: in this question you can work with the vectors in geometric form or convert them to algebraic vectors.

Question 3 (6 points) u, v and w are three coplanar vectors: ⚫ w has a magnitude of 10 and points along the positive x-axis ⚫ v has a magnitude of 3 and makes an angle of 58 degrees to the positive x- axis ⚫ u has a magnitude of 5 and makes an angle of 119 degrees to the positive x- axis ⚫ vector v is located in between u and w a) Draw a diagram of the three vectors placed tail-to-tail at the origin of an x-y plane. b) If possible, find w × (u + v) Support your answer mathematically or a with a written explanation. c) If possible, find v. (ū⋅ w) Support your answer mathematically or a with a written explanation. d) If possible, find u (v × w) Support your answer mathematically or a with a written explanation. Note: in this question you can work with the vectors in geometric form or convert them to algebraic vectors.

Chapter 11 Solutions

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

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Find whether the given vectors u , v , and w are perpendicular.

Find whether the given vectors u, v, and w are perpendicular.

Find whether the given vectors u, v, w, and r are parallel.

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Chapter 11 Solutions