# Math: Vectors

posted by .

I forgot how to do the following, so I'm not sure if they're correct or not:

Find the work done by a force "F" that causes a displacement "d".

i and j = unit vectors

a)
F = 4i + j
d = 3i + 10j

W = (4i + j)(3i + 10j)
W = 12|i|^2 + 40ij + 3ij + 10|j|^2
W = 12(1) + 40(0) + 3(0) + 10(1)
W = 22

b)
F = 2i
d = 5i + 6j

W = (2i)(5i + 6j)
W = 10|i|^2 + 12ij
W = 10(1) + 12(0)
W = 10

• Math: Vectors -

You want F DOT D (I am using capital for vector) = |F||D| cos T
where T is the angle between them.

dot product = scalar product = FxDx + FyDy
= 4*3 + 1*10 = 12+10 = 22

Now do that one other way
|F| = sqrt(4^2+1^2) = sqrt(17)
|D| = sqrt(3^2+10^2) = sqrt (109)
angle F to x axis = tan^-1 (1/4) =14 deg
angle D to x axis = tan^-1 (10/3)=73.3 deg
so T = 73.3 - 14 = 59.3
cosT = .5105
so
W = |F||D| cos T = 21.97 = 22 to our sig figs
check

• Math: Vectors -

b)
F = 2i
d = 5i + 6j

W = 2*5 + 0*6
W = 10

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