Section 13.4
The Cross Product
Torque
• Torque is a measure of how much a force
acting on an object causes that object to rotate
– The object rotates around an axis (the pivot point)
– It depends on the force in the direction of
movement and the distance the force is from the
pivot point
• We can represent the force and the distance as
vectors
– The force vector is in the direction of the applied
force
– The distance vector points away from the pivot
point
Torque
• Torque depends on the force in the direction
of movement and the distance the force is from
the pivot point
F
F
r
pivot point

Torque
  r  F   r  F sin 
F
F
r
pivot point

Geometric
Definition
 
• If a , b are
vectors then their cross product,


a  b is a vector with the following
properties

– Magnitude
 
 
a  b  a b sin 

b

a
– Direction:
Perpendicular to the plane created by the given
vectors
• We can use the “right hand rule” to determine the
direction of the vector
• Contrast this with the dot product which produces a
scalar quantity
Algebraic Definition


• Let a  a1 , a2 , a3  and b  b1 , b2 , b3 
 
then a  b  a2b3  a3b2 , a3b1  a1b3 , a1b2  a2b1 
• This can also be done by computing a 3x3
determinant
i
j
a  b  a1 a2
b1 b2
k
a3 
b3
(a2b3  a3b2 )i  (a1b3  a3b1 ) j  (a1b2  a2b1 )k
Applications of the Cross Product
• Area of a parallelogram

b

h

a
• Recall area of a parallelogram is base x height
• How does the cross product apply?
 
A  a b
Applications of the Cross Product
• What about the area of the triangle formed by
two vectors (starting at the same point)?

b


a
1  
A  a b
2
Applications of the Cross Product
• Planes in 3 space
– Fact: any 3 non-collinear points in 3 space determines
a plane
• How can we determine the unique plane through
these 3 non-collinear points, P, Q, and R?
– Recall: If we have a normal vector to a plane and a
point in the plane, we can determine the equation of a
plane
• We need to find a normal
vector
to
the
plane



n  PQ PR
• Now use point normal form for a plane to find
your equation
• So why do we need both types of vector
multiplication?
• Let’s compare and contrast
• Both operations need two vectors
• The dot product produces a scalar, the cross
product a vector
• What happens in each case if the two vectors
are perpendicular?
– Dot product is zero
– Cross product is maximum
• The dot product can be used to project one
vector on to another
• The cross product can be used to find a vector
that is perpendicular to two given vectors
• The dot product generalizes to any dimension
• The cross product only exists in 3-space
• Both involve the lengths of the two input
vectors and the angle between them