Cross Product

Pre-requisites: Vectors

The cross product, like the dot product, is a product of two vectors which has two definitions. The geometric definition of the cross product is that

v × w = |v| |w| |sin theta| [where once again theta is the angle between the two vectors] and that the direction of the cross product is orthogonal to both v and w. From this definition, it should be clear that the cross product of two vectors IS A VECTOR and not a scalar. Repeat that phrase to yourself a couple of times: the cross product is a vector. Taunt people who believe otherwise.

The definition of the cross product I just gave you is incomplete: there are two vectors that satisfy being orthogonal to v and w and also having the desired magnitude. To pick the correct one, we must use the right hand rule: if you point the thumb of your right hand along v and point your pointing finger along w, then v × w will point in the direction of your middle finger. To put it another way: if v and w are rotated to lie in the part of the xy-plane with y >= 0, then v × w will point in the direction of the positive z-axis.

The algebraic definition of the cross product is

(v₁, v₂, v₃) × (w₁, w₂, w₃) = ( v₂ w₃ - v₃ w₂, v₃ w₁ - v₁ w₃, v₁ w₂ - v₂ w₁ ) This definition is easier to remember if you think of it as the determinant of the matrix

[  i   j   k   ]
[  v₁  v₂  v₃  ]
[  w₁  w₂  w₃  ]

In fact, to some people the whole point of the cross product is that the triple product (what you get when you combine the dot product and the cross product) is the same as a determinant:

                    [  a₁  a₂  a₃  ]
a · ( b × c ) = det [  b₁  b₂  b₃  ]
                    [  c₁  c₂  c₃  ]

If you don't know what a determinant is, you can just use the above equation as a definition and not worry about it. It's most important property is that if you have a parallelopiped (the three dimensional version of the parallelogram -- a solid with eight pairwise parallel faces) with vertices 0, v, w, x, v + w, v + x, w + x, and v + w + x has volume | v · ( w × x ) |. Thus the triple product / determinant of a three by three matrix can be thought of as an "oriented" (signed) volume.

Now, thinking of the cross product in terms of the determinant of a three by three matrix only helps you if you can remember the determinant of a three by three matrix. One way of doing this is to think of it in terms of the determinants of two by two matrices:

    [ i  j  k  ]
det [ v₁ v₂ v₃ ] = det [ v₂ v₃ ] - det [ v₁ v₃ ] + det [ v₁ v₂ ] 
    [ w₁ w₂ w₃ ]       [ v₂ v₃ ]       [ v₁ v₃ ]       [ v₁ v₂ ]

where

det [ a b ] = ad - bc = (a,b) · (d,-c)
    [ c d ]

The determinant of a two by two matrix also has a geometric interpretation: it is the area of the parallelogram with vertices 0, v, w, and v + w .

Wow. We've made quite a number of strong statements, without giving a hint of proof to any of them. Let's rectify this situation:

Statement: The algebraic and geometric definition of the cross product are the same.
Proof: First, the algrebraic definition and the geometric definition of the cross product agree in magnitude:
sqrt( ( v₂ w₃ - v₃ w₂ )² + ( v₃ w₁ - v₁ w₃ )² + ( v₁ w₂ - v₂ w₁ )² ) = sqrt( ( v₁v² + v₂v² + v₃v² ) ( w₁v² + w₂v² + w₃v² ) - ( v₁ w₁ + v₂ w₂ + v₃ w₃ )² )
= sqrt( | v |² | w |² - ( v · w )² ) = sqrt( | v |² | w |² - | v |² | w |² cos² theta ) = | v | | w | cos theta Next, the algebraic definition of v × w is orthogonal to both v and w: ( v₂ w₃ - v₃ w₂, v₃ w₁ - v₁ w₃, v₁ w₂ - v₂ w₁ ) · ( v₁, v₂, v₃ ) = v₂ w₃ v₁ - v₃ w₂ v₁ + v₃ w₁ v₂ - v₁ w₃ v₂ + v₁ w₂ v₃ - v₂ w₁ v₃ = 0 and similarly for ( v × w ) · w because w × v = - v × w .
Finally, the algebraic definition of the cross product obeys the right hand rule because it does for (c, 0, 0) × (a,b,0) = (0, 0, cb ) [which is "up" if (a,b,0) is on the positive y side of the plane, and "down" if (a,b,0) if on the negative y side of the plane assuming that c > 0], and the general case of v × w can always be smoothly rotated into that case. The right hand rule is unchanged by rotations, and the algebraic definition of the cross product changes continuously with smooth rotations so if it obeys it before a smooth rotation, it must obey it afterwards too. Q.E.D.
Statement: The magnitude of the cross product gives the area of the parallelogram with vertices .
Proof: Trigonometry.
Corollary: The determinant of the two by two matrix gives the area of the parallelogram with vertices (0,0), (a,b), (c,d), and (a+c,b+d)..
Proof: = (a,b,0) × (c,d,0)
Statement: The absolute value of the triple product a · ( b × c ) gives the volume of the parallelopiped with vertices 0, v, w, x, v + w, v + x, w + x, and v + w + x .
Proof: a · ( b × c ) which is the base of the parallelopiped times its height.

If you found the above proofs a little abstract, you may find it useful to interactively play around with the cross product to develop your intuitions. You may also take comfort in the fact that the following simple properties are all that you really need to know about the cross product:

Anticommutativity: w × v = - v × w . [I know that I said it already before, but it is worth repeating.]
Linearity: v × ( w + x ) = v × w + v × x . Also, ( c v ) × w = c ( v × w ) and 0 × v = 0.
Idempotence: v × v = 0.
Cyclicity: i × j = k, j × k = i, k × i = j.

Finally, here's an application of the cross product: finding the equation of a plane given two vectors and a point lying on the plane. We did this before by solving a system of linear equations, but it is much simpler if you use the cross product. Just take the cross product of the two vectors to get a vector orthogonal to both of them and thus a normal vector to the plane, and then plug the normal vector (A,B,C) and the point (x₁, x₂, x₃) into the formula A ( x - x₁ ) + B ( y - x₂ ) + C ( z - x₃ ) = 0

Exercises:

What do you get when you cross a grape and an elephant? When you cross a grape and a mountain climber?
Is the cross product associative?
Calculate the cross product of (1,2,3) and (4,5,6).
Prove that the cross product is linear in its second argument.
Prove that the cross product factors constants in its second argument.
What is i × k? j × i?
Find a vector orthogonal to (1,2,3) and (0,4,-9).
Find the volume of the parallelopiped with the vertices .
Find the equation of the plane going through the points (0,0,0), (1,2,3), and (2,0,-1).
Find the equation of the plane going through (1,2,3) and containing the vectors (0,0,1) and (2,-3,7).
Prove that the cross product of two vectors is the zero vector if and only if the two vectors are parallel or one of them is the zero vector.
Prove the Jacobi Identity:
Show that determinants can factor a scalar from a row or column.
Show that a determinant doesn't change if one row (or column) is added to another row (or column).
Show that the system of equations has a solution if and only if the determinant isn't zero.
Show that switching two rows (or columns) changes the sign of a determinant.

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last modified 1 July 1997