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.
\( W = \| proj_{\vec{d}} \vec{F} \| \| \vec{d} \| \)
\( W = \vec{F} \cdot \vec{d} \)
Recommended Books on Amazon (affiliate links) | ||
---|---|---|
![]() |
![]() |
![]() |
Really UNDERSTAND Calculus
Log in to rate this page and to see it's current rating.
external links you may find helpful |
---|
To bookmark this page and practice problems, log in to your account or set up a free account.
Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.
| |
Join Amazon Prime - Watch Thousands of Movies & TV Shows Anytime - Start Free Trial Now |
---|
I recently started a Patreon account to help defray the expenses associated with this site. To keep this site free, please consider supporting me. |
---|
Support 17Calculus on Patreon |