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Cross Product
Ali Tamaki
Ben Waters
Linear Systems
Spring 2006
Definition
The cross product is defined as the vector perpendicular to
two given vectors A and B separated by an angle  and is
shown by:
||A x B|| = ||A|| ||B|| sin
The magnitude of the cross product equals the area of the
parallelogram that the initial two vectors span
Properties
Anti-commutative:
a x b = -b x a
Distributive over addition:
a x (b + c) = a x b + a x c
Compatible with scalar multiplication:
(ra) x b = a x (rb) = r(a x b)
Not associative, but satisfies the Jacobi identity:
a x (b x c) + b x (c x a) + c x (a x b) = 0
Cross product is only valid in R3 and R7
Solving Methods
Given two vectors u = [u1 u2 u3] and v = [v1 v2 v3]
u x v = u2v3 – u3v2
u3v1 – u1v3
u1v2 – u2v1
Matrix Solving Method
The first row is the standard basis vectors and must appear in the order
given here. The second row is the components of u [u1 u2 u3] and the
third row is the components of v [v1 v2 v3].
Given two vectors u and v, u x v equals the determinant of the matrix.
Applications
Mathematical Applications
Vector Cross Product
Geometry
Physics Applications
Torque
Angular Momentum
Lorentz Force
Vector Cross Product
Cross Product Applet
||a x b|| = ||a|| ||b|| sin()
Two non-zero vectors a and b are parallel iff a x b = 0
Geometric Applications
3-D
Volume of a parallelepiped with sides of length a, b, c
equals the magnitude of the scalar triple product:
V = |a · (b x c)|
Torque & Angular Momentum
Torque is the measure of how much a force acting on
an object will cause that object to rotate. Ø is
T= r x F = r F sin(ø)
If a particle with linear momentum p is at a position r with
respect to some point, then its angular momentum L is the
cross product of r and p
L=rxp
Torque Example
T= R x F = RFsin()
 is the angle between the
location of the applied force and
the point where the radius meets
this force.
In this case,  = 90, which
means that the torque is just the
product of the radius and the
applied force because sin(90) =
1.
F
R
Lorentz Force
The Lorentz Force F exerted on a charged particle in an
electromagnetic field equals the electric charge q of the
particle times the electric field E plus the cross product of
the velocity v of the particle and the magnetic field B
F = q (E + v x B)
Lorentz Force RHR
Align the thumb of the right
hand in the direction of the
velocity v. Then, point the
index finger of the right
hand in the direction of the
magnetic field B. Now, the
palm of the right hand points
in the direction of the
Lorentz force F.
Lorentz Force Applet
Conclusion
What do you get when you cross a mountain-climber with
a mosquito?
Nothing, you can't cross a scalar with a vector.