3. (10 pts) Compute the torque on an object if a force of ~
F=−3ˆı+ 7 ˆ−4ˆ
kN is exerted at the position
~r = 6ˆı−10ˆ+ 9ˆ
km.
You can either multiply out all 9 terms and then work the cross-products of the unit vectors or use the
determinant method. Both techniques give the same answer.
First, the direct multiplication method:
~τ =6ˆı−10 ˆ+ 9 ˆ
k×−3ˆı+ 7 ˆ−4ˆ
k
=−18 (ˆı׈ı)+ 42 (ˆı׈)−24 ˆı׈
k+30 (ˆ×ˆı)−70 (ˆ×ˆ) +40 ˆ×ˆ
k−27 ˆ
k׈ı+ 63 ˆ
k׈−36 ˆ
k׈
k
= 4 2 ˆ
k+ 24 ˆ−30 ˆ
k+ 40ˆı−27 ˆ−63 ˆı
~τ =−23 ˆı−3 ˆ+ 12 ˆ
kN·m
And then the determinant method:
~τ =
ˆıˆˆ
k
6−10 9
−3 7 −4
= ˆı(40 −63) −ˆ(−24 + 27) + ˆ
k(42 −30)
~τ =−23 ˆı−3 ˆ+ 12 ˆ
kN·m
There was a lot of very sloppy notation used in this problem. The most prevalent was a lack of vector
signs, like:
τ=r×F
which is not correct! The magnitude of the torque is not equal to the product of the magnitudes of r
and F. You must also multiply by the sin of the angle between them! However, with the vector symbols
it is correct,
~τ =~r ×~
F
Also, there were a lot of vector components written with i,j, and krather than ˆı,ˆ, or ˆ
k.i,j, and kare
just (scalar) variables unless you specifically indicate, with the hat symbol, that they are unit vectors.
Another notational mistake occurred in the actual vector multiplication. Torque is a vector, so the type
of product must have a vector as a result. Therefore it cannot be the result of a dot product! So
~τ =~r ·~
F
is completely wrong. You could compute the dot product of ~r and ~
F, but it does not give anything
related to the torque. I also saw the expression written as
~τ =~r ~
F
or even with the components of the two specified vectors,
~τ =6 ˆı−10 ˆ+ 9 ˆ
k −3ˆı+ 7 ˆ−4ˆ
k
(with or without appropriate hats on the unit vectors) but with no multiplication symbol. For multiply-
ing vectors, you must specify which multiplication you intend to use, otherwise it does not make sense.
~
A~
Bis not defined. You can’t do it.
While I did not subtract any points for these notational errors, I did take off points if the order of the
vectors in the cross product was reversed, ~
F×~r, since this gives −~τ rather than ~τ .