Question
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2. A conveyor belt generates a force \( \vec{F}=5 \mathbf{i}-3 \mathbf{j}+3 \mathbf{k} \) that moves a suitcase from point \( \mathrm{P}(1,1,1) \) to Point \( \mathrm{Q}(9,4,7) \) along a straight line. Find the work done by the conveyor belt. The distance is measured in meters and the force is measured in newtons. 3. If \( \mathbf{u} \) and \( \mathbf{v} \) are vectors represented as \( 2 \mathbf{i}+3 \mathbf{j}+4 \mathbf{k} \) and \( \mathbf{i}+2 \mathbf{j}+3 \mathbf{k} \), then find the area of the parallelogram spanned between them. Find the area of the triangle whose vertices are \( \mathrm{A}(3,-1,2), \mathrm{B}(1,-1,-3) \) and \( \mathrm{C}(4,-3,1) \). 5. Find the area of the quadrilateral whose vertices are \( \mathrm{A}(1,0), \mathrm{B}(3,5), \mathrm{C}(7,2) \) and \( \mathrm{D}(5,-2) \). A car mechanic applies a force of 800 N to a wrench to loosen a bolt. She applies the force perpendicular to the arm of the wrench. The distance from the bolt to her hand is 0.40 m . What is the magnitude of the torque applied?

Ask by Kelly Matthews. in Ethiopia
Mar 09,2025

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Answer

1. The work done by the conveyor belt is 49 Joules. 2. The area of the parallelogram spanned by vectors **u** and **v** is √6. 3. The area of the triangle with vertices A(3,-1,2), B(1,-1,-3), and C(4,-3,1) is (1/2)√165. 4. The area of the quadrilateral with vertices A(1,0), B(3,5), C(7,2), and D(5,-2) is 23. 5. The magnitude of the torque applied by the mechanic is 320 N·m.

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The Deep Dive

Did you know that the first known use of a torque wrench dates back to the 1910s? It revolutionized how mechanics applied force, ensuring bolts are secured to the correct specifications without risking damage. This historical advancement made today's vehicles safer and easier to maintain! When tackling problems involving force and torque, a common mistake is forgetting the torque's direction. Remember, torque is not just about the magnitude but also involves rotational direction—clockwise or counterclockwise can influence mechanical outcomes significantly. Always apply the right-hand rule to visualize the direction of your torque!

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