Applications of Vector Dot & Cross Products


Work (in physics) is done whenever a force causes linear displacement. That is to say, work is the amount of force along the displacement vector.

Torque is the turning effect done by a force that causes angular displacement.

Work
\( W = \vec{F}\cdot\vec{d} \)
\( W = \vert\vec{F}\vert\vert\vec{d}\vert\cos{\theta} \)
Unit is \( N\cdot m \) or Joules
Area of llgm
\( Area = l \cdot w \)
\( Area = \vert\vec{a}\vert\vert\vec{b}\vert\sin{\theta} \)
\( Area = \vert\vec{a} \times \vec{b}\vert \)
Torque
\( \vec{T} = \vec{r} \times \vec{F} \)
\( \vec{T} = \vert\vec{r}\vert\vert\vec{F}\vert\sin{\theta} \text{ } \hat{n} \)
Unit is \( N\cdot m \) or Joules

Find the area of the triangle with the vertices P(7,2,-5), Q(9,-1,-6), and R(7,3,-3).


A 25kg box is located 8m up a ramp inclined at an angle of 18° to the horizontal. Determine the work done by the force of gravity as the box slides to the bottom of the ramp.


Find the torque produced by a cyclist exterting a force of 115 N on a pedal in the position shown in the diagram, if the shaft of the pedal is 16cm long.




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