Vector Multiplication: Cross Product

We define the cross product of two three-dimensional vectors a and b by the requirements:

• a × b is a vector that is perpendicular to both a and b.
• ||a × b|| is the area of the parallelogram spanned by a and b (i.e. the parallelogram whose adjacent sides are the vectors a and b).
• The direction of a×b is determined by the right-hand rule. (This means that if we curl the fingers of the right hand from a to b, then the thumb points in the direction of a × b.)

From simple trigonometry, we can calculate that the area of the parallelogram spanned by a and b is

||a|| ||b|| sin θ,

where θ is the angle between a and b. (One can view the parallelogram as having a base of length ||b|| and perpendicular height ||a|| sin θ, as pictured below.)

This formula shows that the magnitude of the cross product is largest when a and b are perpendicular. On the other hand, if a and b are parallel or if either vector is the zero vector, then the cross product is the zero vector. (It is a good thing that we get the zero vector so that the above definition still makes sense. If the vectors are parallel or one vector is the zero vector, then there is not a unique line perpendicular to both a andb. But since there is only one vector of zero length, the definition still uniquely determines the cross product.)

Cross product of unit vectors

Let i, j, and k be the standard unit vectors in R³. (We define the cross product only in three dimensions.)

The parallelogram spanned by any two of these standard unit vectors is a unit square, which has area one. Hence, by the geometric definition, the cross product must be a unit vector. Since the cross product must be perpendicular to the two unit vectors, it must be equal to the other unit vector or the opposite of that unit vector. Looking at the above graph, you can use the right-hand rule to determine the following results.

i × j	= k
j × k	= i
k × i	= j

This little cycle diagram can help you remember these results.

What about i × k? By the right-hand rule, it must be -j. I think you get the idea (after all b × a = -a × b).

j × i	= -k
k × j	= -i
i × k	= -j

Finally, the cross product of any vector with itself is the zero vector (a × a = 0). So the cross product of any standard unit vector with itself the zero vector.

General vectors

With the exception of the two special properties mentioned above (b×a = -a×b, and a×a = 0), the cross product behaves like regular multiplication. It obeys the following properties:

• (ya) × b = y(a × b) = a × (yb),
• a × (b + c) = a × b + a × c,
• (b + c) × a = b × a + c × a,

where a, b, and c are vectors in R³ and y is a scalar. We can use these properties, along with the cross product of the standard unit vectors, to write the formula for the cross product in terms of components.

We write the components of a and b as:

a = (a1,a2,a3) = a1i + a2j + a3k
b = (b1,b2,b3) = b1i + b2j + b3k

First, we’ll assume that a₃ = b₃ = 0. (Then, the manipulations are much easier.) We calculate:

a × b	= (a1i + a2j) × (b1i + b2j)
	= a1b1(i × i) + a1b2(i × j) + a2b1(j × i) + a2b2(j × j)

Since we know that i × i = 0 = j × j and that i × j = k = -j × i, this quickly simplifies to

a × b	= (a1b2 - a2b1)k
	= k.

Writing the result as a determinant, as we did in the last step, is a handy way to remember the result.

What about the general case where a₃ and b₃ might not be zero? Well, that’s too many manipulations for me to write it out here, and you probably wouldn’t read it anyway. To calculate the result, it’s just the same process as above. You can verify yourself that in general

a × b	= (a1i + a2j + a3k) × (b1i + b2j + b3k)
	= i -j + k

Looking at the formula for the 3 × 3 determinant, we see that the formula for a cross product looks a lot like the formula for the 3 × 3 determinant. If we allow a matrix to have the vector i, j, and k as entries (OK, maybe this doesn’t make sense, but this is just as a tool to remember the cross product), the 3 × 3 determinant gives a handy mnemonic to remember the cross product:

a × b =

Source: math.umn.edu