Determine the moment about the origin O of the force F=4i-3j+5k that acts at a point A. Assume that the position vector of A is: (a) r=2i+3j-4k (b) r=-8i+6j-10k (c) r=8i-6j+5k Hint (in order): M_(O)=3i-26j-18k M_(O)=-15i-20j

To determine the moment about the origin O of the force F that acts at a point A, we can use the formula:<br /><br />$M_O = r \times F$<br /><br />where M_O is the moment about the origin O, r is the position vector of point A, and F is the force vector.<br /><br />Let's calculate the moment for each given position vector:<br /><br />(a) $r = 2i + 3j - 4k$<br /><br />$M_O = r \times F = (2i + 3j - 4k) \times (4i - 3j + 5k)$<br /><br />Using the cross product formula, we get:<br /><br />$M_O = (2 \cdot (-3) - 3 \cdot 5)i - (4 \cdot 5 - (-4) \cdot 4)j + (4 \cdot 3 - 2 \cdot (-3))k$<br /><br />$M_O = (-6 - 15)i - (20 + 16)j + (12 + 6)k$<br /><br />$M_O = -21i - 36j + 18k$<br /><br />Therefore, the moment about the origin O of the force F that acts at point A is $M_O = -21i - 36j + 18k$.<br /><br />(b) $r = -8i + 6j - 10k$<br /><br />$M_O = r \times F = (-8i + 6j - 10k) \times (4i - 3j + 5k)$<br /><br />Using the cross product formula, we get:<br /><br />$M_O = (-8 \cdot (-3) - 6 \cdot 5)i - (4 \cdot 5 - (-10) \cdot 4)j + (4 \cdot 6 - (-8) \cdot (-3))k$<br /><br />$M_O = (24 - 30)i - (20 + 40)j + (24 - 24)k$<br /><br />$M_O = -6i - 60j + 0k$<br /><br />Therefore, the moment about the origin O of the force F that acts at point A is $M_O = -6i - 60j$.<br /><br />(c) $r = 8i - 6j + 5k$<br /><br />$M_O = r \times F = (8i - 6j + 5k) \times (4i - 3j + 5k)$<br /><br />Using the cross product formula, we get:<br /><br />$M_O = (8 \cdot (-3) - (-6) \cdot 5)i - (4 \cdot 5 - 5 \cdot 8)j + (4 \cdot (-6) - 8 \cdot (-3))k$<br /><br />$M_O = (-24 + 30)i - (20 - 64)j + (-24 + 24)k$<br /><br />$M_O = 6i + 44j + 0k$<br /><br />Therefore, the moment about the origin O of the force F that acts at point A is $M_O = 6i + 44j$.