Moment - Vector Product
You have learned in the past basic operations for numbers such as addition,
subtraction, multiplication and division. These operations allowed you to do useful things
with numbers. As we have previously discussed, vectors are different entities and the
rules/operations learned for normal numbers no longer apply. Although some operations
such as addition have the same name, the operation itself is carried out in a different way
(i.e. head to tail method). Likewise there are operations for vectors that have no
counterpart in ordinary numbers. One of these is used to find the moment of a force about
a point.
The two vectors P and Q shown form a plane. The angle between their directions is θ
o
(0 < θ < 180 ).
V
Q
θ
P
The vector product P × Q produces a new vector V ( V = P × Q ) whose magnitude and
direction are given by
| V | = PQsinθ
direction: perpendicular to the plane of P and Q
The perpendicular to a plane can be in either of two opposing directions and the correct
one as shown above is given by the right hand rule.
Cross Product Properties
The cross product exists only for vectors and does not exist for ordinary numbers.
The cross product satisfies the following rules.
(P×Q)=-(Q×P)
P×(Q1+Q2)= P×Q1+ P×Q2
(P×Q)×S ≠ P×(Q×S)
It is very important to note the negative sign in the first rule. If you switch the order, the
resulting vector will point in the opposite direction.
Unit Vectors
The cross product for unit vectors produces a new unit vector perpendicular to the
ones crossed. For i, j and k, this can be remembered by the figure
k
i
j
That is to say if you move counter clockwise i × j = k or k × i = j while if you move
clockwise the result is negative (j × i = -k or i × k = -j ).
Rectangular Components
In some cases the vectors P and Q are given in rectangular components as
P = Px i + Py j + P z k
Q = Qx i + Qy j + Qz k
In this case you can work out the cross product term by term as
V = P × Q = ( Px i + Py j + Pz k ) × ( Qx i + Qy j + Qz k )
= i ( PyQz – PzQy ) + j ( PzQx – PxQz ) + k ( PxQy – PyQx )
The above formula for V can also be obtained from the 3 × 3 determinant
V
i
j
k
Px
Py
Pz
Qx
Qy
Qz
Moment of a Force
Vector (or cross) product is useful for finding the moment of a force about a point.
Moment is a vector. The figure below shows the moment vector M o of force F about
point O.
Mo
F
O
d
A
r
B
The magnitude of the moment of F about O is defined as M o = Fd, where d is the shortest
distance from O to the line of the force. The direction of M o is perpendicular to the
plane formed by the line of F and point O. The vector M o can be found from the cross
product
Mo=r×F
where r is a vector from point O to any point (choose a convenient one) along the line of
F. The correct direction of M o can be found by the right hand rule.