## 17Calculus - Dot Product Application - Work

##### 17Calculus

One of the major applications of the dot product is to calculate work.

Work is defined as the magnitude of a force acting on an object times the distance the object moves. Force is a vector and the only part of the vector that contributes to the work is the part in the direction the object moves. So, if we define a vector $$\vec{d}$$ that points in the direction that the object moves whose magnitude is the distance moved from point $$P$$ to point $$Q$$ (see the figure to the right), the work, $$W$$, is

$$W = \| proj_{\vec{d}} \vec{F} \| \| \vec{d} \|$$

Now, the length of the projection of vector $$\vec{F}$$ onto vector $$\vec{d}$$ is $$\| proj_{\vec{d}} \vec{F} \| = \|\vec{F}\| \cos(\theta)$$
So the work equation becomes $$W = \|\vec{F}\| \cos(\theta) \| \vec{d} \| = \vec{F} \cdot \vec{d}$$.

Here we have shown there are two (equivalent) equations to calculate the work.

$$W = \| proj_{\vec{d}} \vec{F} \| \| \vec{d} \|$$

$$W = \vec{F} \cdot \vec{d}$$

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