Explanation Given: The force in the form of Cartesian vector , F = { − 6 i + 9 j + 3 k } . Draw the free body diagram of forces F and line AO as in Figure (1). Let us establish a position vector for points A and O . The coordinate points A ( 1 , − 2 , 2 ) m and point O ( 0 , 0 , 0 ) m. Express the position vector r in the direction of AO using Figure (1). r A O = ( x O − x A ) i + ( y O − y A ) j + ( z O − z A ) k (I) 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 AO . r A O = ( x O − x A ) 2 + ( y O − y A ) 2 + ( z O − z A ) 2 (II) Express the force in Cartesian vector form. F = { − 6 i + 9 j + 3 k } Express the magnitude of the direction of F . F = ( − 6 ) 2 + ( 9 ) 2 + ( 3 ) 2 = 36 + 81 + 9 = 11.22 kN Determine the unit vector of AO . u A O = ( r A O r A O ) (III) Apply the dot product to find the angle between F and line AO . u A O ⋅ F = r A O ⋅ F cos θ θ = cos − 1 ( u A O ⋅ F ) F (IV) Conclusion: Substitute 1 m for x A , 2 m for y A , 2 m for z A , 0 for x O , 0 for y O , and 0 m for z O in Equation (I). r A O = ( 0 − 1 ) i + ( 0 − ( − 2 ) ) j + ( 0 − 2 ) k = − i + 2 j − 2 k Substitute 1 m for x A , 2 m for y A , 2 m for z A , 0 for x O , 0 for y O , and 0 m for z O in Equation (II)

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The area of science and mathematics concerned the motion of the objects, the relation between the force matter, and motion if specified. Forces exerted over the objects result in displacements or changes in the position of an object concerning its enviro…

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Chapter 2.9, Problem 25FP

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The angle θ between the force and the line AO.

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The angle θ between the force and the line AO .

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