Explanation Establish a position vector for points O , A , B , and C . The coordinates of the points are: O ( 0 , 0 , 0 ) m , A ( 0 , 4 , 3 ) m , B ( 1.5 , 0 , 6 ) m , and C ( − 2 , 0 , 4 ) m , respectively. Express the position vector r in the direction of AB . r A B = ( x B − x A ) i + ( y B − y A ) j + ( z B − z A ) k (I) Here, the coordinates of A and B are x A , x B , y A , y B , z A , and z B , respectively. Express the position vector r in the direction of AC . r A C = ( x C − x A ) i + ( y C − y A ) j + ( z C − z A ) k (II) Here, the coordinates of A and C are x A , x C , y A , y C , z A , and z C , respectively. Express the position vector r in the direction of AO . r A O = ( x O − x A ) i + ( y O − y A ) j + ( z O − z A ) k (III) Here, the coordinates of A and O are x A , x O , y A , y O , z A , and z O , respectively. Express the magnitude of the direction of AB . r A B = ( x B − x A ) 2 + ( y B − y A ) 2 + ( z B − z A ) 2 (IV) Express the magnitude of the direction of AC . r A C = ( x C − x A ) 2 + ( y C − y A ) 2 + ( z C − z A ) 2 (V) Express the magnitude of the direction of AO . r A O = ( x O − x A ) 2 + ( y O − y A ) 2 + ( z O − z A ) 2 (VI) Determine the unit vector of AB . u A B = ( r A B r A B ) (VII) Here, the position vector in the direction of AB is r A B , and the magnitude of the position vector in the direction of AB is r A B . Determine the unit vector of AC . u A C = ( r A C r A C ) (VIII) Here, the position vector in the direction of AC is r A C , and the magnitude of the position vector in the direction of AC is r A C . Determine the unit vector of AO . u A O = ( r A O r A O ) (IX) Here, the position vector in the direction of AO is r A O , and the magnitude of the position vector in the direction of AO is r A O . Apply the dot product to find the angle θ . u A B ⋅ u A O = cos θ θ = cos − 1 ( u A B ⋅ u A O ) (X) Apply the dot product to find the angle ϕ . u A C ⋅ u A O = cos θ θ = cos − 1 ( u A C ⋅ u A O ) (XI) Conclusion: Substitute 0 for x A , 4 m for y A , 3 m for z A , 1.5 m for x B , 0 for y B , and 6 m for z B in Equation (I). r A B = ( 1.5 − 0 ) i + ( 0 − 4 ) j + ( 6 − 3 ) k = ( 1.5 i − 4 j + 3 k ) m Substitute 2 m for x C , 0 for y C , 4 m for z C , 0 for x A , 4 m for y A , and 3 m for z A in Equation (II). r A C = ( − 2 − 0 ) i + ( 0 − 4 ) j + ( 4 − 3 ) k = ( − 2 i − 4 j + 1 k ) m Substitute 0 for x A , 4 m for y A , 3 m for z A , 0 m for x O , 0 for y O , and 0 m for z O in Equation (III)

Engineering Mechanics: Statics & Dynamics (14th Edition)

14th Edition

ISBN: 9780133915426

Author: Russell C. Hibbeler

Publisher: PEARSON

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Chapter 2.9, Problem 130P

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The angle θ and ϕ between the axes OA of the flag pole and AB and AC respectively.

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Chapter 2.9, Problem 129P

Chapter 2.9, Problem 131P

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Engineering Mechanics: Statics & Dynamics (14th Edition)

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The angle θ and ϕ between the axes OA of the flag pole and AB and AC respectively.

Solution Summary: The author determines the angle between the axes OA of the flag pole and AB and AC. Establish a position vector for points O, A, B, and C.

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The angle θ and ϕ between the axes OA of the flag pole and AB and AC respectively.

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