The Vector Product

A product that is a vector

The dot product of two vectors is a number. The vector product of two vectors in three dimensions is a new vector, written a × b and read “a cross b”, which is why it is also called the cross product.

For $a = \left(a_{1}, a_{2}, a_{3}\right)$ and $b = \left(b_{1}, b_{2}, b_{3}\right)$, it is $a \times b = \left(a_{2}b_{3} - a_{3}b_{2}, a_{3}b_{1} - a_{1}b_{3}, a_{1}b_{2} - a_{2}b_{1}\right)$.

Each component leaves out its own place and uses the other two, taken round the cycle 1, 2, 3, 1. For the first component, start at place 2: $a_{2}b_{3} - a_{3}b_{2}$. For the second, start at place 3: $a_{3}b_{1} - a_{1}b_{3}$. For the third, start at place 1: $a_{1}b_{2} - a_{2}b_{1}$. Each time, a comes from the starting place and b from the next place round, and the product taken away swaps those two places.

Two flat arrows

Take a = (2, 1, 0) and b = (1, 3, 0). Both have a third component of 0, so both lie flat in the floor, the plane z = 0.

The first component is $a_{2}b_{3} - a_{3}b_{2} = 1 \times 0 - 0 \times 3 = 0$. The second is $a_{3}b_{1} - a_{1}b_{3} = 0 \times 1 - 2 \times 0 = 0$. The third is $a_{1}b_{2} - a_{2}b_{1} = 2 \times 3 - 1 \times 1 = 6 - 1 = 5$. So a × b = (0, 0, 5): no part along the floor at all, and 5 straight up.

a = (2, 1, 0) and b = (1, 3, 0), seen from above. They meet at 45°, and a × b = (0, 0, 5) points straight up out of the page. Its length 5 is $|a| |b| \sin 45^\circ = \sqrt{5} \times \sqrt{10} \times \frac{1}{\sqrt{2}} = 5$.

a and b lie in the shaded plane, with lengths 3 and 2 and an angle of 60° between them. a × b stands at right angles to the plane, with length 3 × 2 × sin 60° = 5.20 to 2 decimal places. Swing b round: past 180° the product points down instead, and when b lies along the line of a, at 180° or 360°, the product is zero.

At right angles to both

The point of the vector product is its direction: a × b is perpendicular to a and to b. The dot product checks it. For the flat pair, a · (a × b) = 2 × 0 + 1 × 0 + 0 × 5 = 0 and b · (a × b) = 1 × 0 + 3 × 0 + 0 × 5 = 0. Both are zero, so (0, 0, 5) is perpendicular to each.

This holds for every a and b. $a \cdot \left(a \times b\right) = a_{1}\left(a_{2}b_{3} - a_{3}b_{2}\right) + a_{2}\left(a_{3}b_{1} - a_{1}b_{3}\right) + a_{3}\left(a_{1}b_{2} - a_{2}b_{1}\right)$. Multiplied out, that is $a_{1}a_{2}b_{3} - a_{1}a_{3}b_{2} + a_{2}a_{3}b_{1} - a_{1}a_{2}b_{3} + a_{1}a_{3}b_{2} - a_{2}a_{3}b_{1}$, and the six terms cancel in pairs, leaving 0. The same happens with b.

A full example

Take a = (2, −1, 3) and b = (1, 4, −2). The first component is (−1) × (−2) − 3 × 4 = 2 − 12 = −10. The second is 3 × 1 − 2 × (−2) = 3 + 4 = 7. The third is 2 × 4 − (−1) × 1 = 8 + 1 = 9. So a × b = (−10, 7, 9).

Check with dot products: a · (a × b) = −20 − 7 + 27 = 0 and b · (a × b) = −10 + 28 − 18 = 0. A check that does not come out to 0 means a slip, most often a sign in the middle component.

The order matters

Swap the two vectors and each component subtracts the same two products the other way round, so every sign changes: b × a = −(a × b). For the flat pair, (1, 3, 0) × (2, 1, 0) = (0, 0, 1 × 1 − 3 × 2) = (0, 0, −5), the same length pointing straight down.

The right-hand rule gives the direction. Point the fingers of your right hand along a and curl them toward b: your thumb points along a × b. With the unit vectors, i × j = k, j × k = i and k × i = j, while j × i = −k.

Parallel vectors give zero

A vector crossed with itself is the zero vector. In a × a, each component subtracts a product from itself: $a_{2}a_{3} - a_{3}a_{2} = 0$, and the same in the other two places.

The same is true for any two parallel vectors. (2, 1, 0) × (4, 2, 0) = (0, 0, 2 × 2 − 1 × 4) = (0, 0, 0). The length of a × b is $|a| |b| \sin \theta$, and sin 0° = sin 180° = 0, so two vectors along one line have no vector product.

Sums and scalars

The vector product distributes over addition: a × (b + c) = a × b + a × c. Check it with a = (2, 1, 0), b = (1, 3, 0) and c = (3, 1, 0). Then b + c = (4, 4, 0) and a × (b + c) = (0, 0, 2 × 4 − 1 × 4) = (0, 0, 4). Separately, a × b = (0, 0, 5) and a × c = (0, 0, 2 × 1 − 1 × 3) = (0, 0, −1), which add to (0, 0, 4).

A scalar can be taken outside: (2a) × b = 2(a × b). The only rule that does not carry over from ordinary multiplication is the order.

The usual mistakes

Giving a number. The vector product has three components; a single number is the dot product.

Getting the middle component the wrong way round. It is $a_{3}b_{1} - a_{1}b_{3}$, starting from place 3; the dot-product check catches the slip.

Adding the two products. For the flat pair, 2 × 3 + 1 × 1 = 7 is not the third component; it is 2 × 3 − 1 × 1 = 5.

Ignoring the order. b × a is the opposite vector to a × b, not the same one.

The turning effect of a force

In mechanics, a force F acting at a point with position vector r from a pivot has a moment, its turning effect about the pivot, equal to r × F, in that order. The moment points along the axis the force turns about, and its length is the size of the turning effect. In the application below, a bolt holds a sign against the wind.