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Illustration 6.1: Dot Products
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We talk about work as the amount of force in the direction of an object's displacement, Δx, multiplied by the displacement, Δx. No displacement, no work. Work is positive if F and Δx point in the same direction and negative if F and Δx point in opposite directions. This statement is fine if the force and displacement lie on the same axis (in other words lie on a line). What happens if they do not lie on the same axis? When working in two dimensions, the force and the displacement can point in any direction. So how much of a given force is in the direction of the displacement? (We could consider the equivalent description of the amount of the displacement in the direction of the force.)
To answer this question we must use the mathematical construction of the scalar or dot product. The dot product is defined as the scalar product of two vectors: A dot B = A · B = A B cos(θ), where θ is the angle between the two vectors and A and B are the magnitudes of the vectors A and B, respectively. Restart.
Drag the tip of either arrow (position is given in meters). The red arrow is A and the green arrow is B. The magnitude of each arrow is shown and the dot product is calculated. When is the dot product zero? The dot product is zero when the vectors are perpendicular. For any two vectors, the magnitude of the dot product is a maximum when the vectors point in the same (or opposite) directions and a minimum when the vectors are perpendicular. Note also that the assignment of which vector is A and which vector is B does not matter.
So the dot product has the right properties to help us mathematically describe WORK. In general, for a constant force,
WORK = F · Δx = F Δx cos(θ),
where F is the constant force and Δx is the displacement. F and Δx are the magnitude of the vectors, respectively. You may have heard or seen "WORK = Fd" which is not always correct. That statement ignores the vector properties of F and Δx and can lead you into thinking that the definition of WORK is FORCE times DISTANCE, which it is not.