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W = F̅Δs cos θ


W =  work done F =  force exerted Δs =  displacement caused by force θ =  force–displacement angle

when force and displacement are parallel, life is simple.

Apply a force → Displace an object → Do work

When force is not quite parallel to displacement, it's like less force is being used to do the work.

Apply less force → Do less work

That's also pretty simple. When the angle between force and displacement reaches 90°, the component of the force parallel to the displacement reduces to zero.

Apply no force → Do no work

OK, that was counterintuitive at first, but now it seems simple too. The farther the two vectors get from parallel, the less work is done. Expand the angle beyond 90°. Force and displacement are starting to point in opposite directions. At 90°, no work was done. Beyond 90°, less than no work must be done. This is negative work.

Apply less than no force → Do less than no work

There is another reason to embrace negative work. The sign of work indicates the direction of a change. A negative sign indicates a loss of something. In the case of lowering a book, it means lowering its ability to do work — lowering its energy.


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Work = F.s = |Fs| cosα 
where α is the angle between force(F) and displacement(s).

three cases are:
case1 - 0≤α<90
the work is positive as cosα is positive.
case2 - α=90
the work is 0 as cosα = cos 90 = 0
case3 - 90<α≤180
the work is negative as cosα is negative.
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