Given two vectors in three dimensions, one can define their vector or cross product as new vector, perpendicular to both original vectors, and with magnitude proportional to the sine of the angle from the first to the second vector.
In this animation, the cross product of two vectors a (blue) and b (red) are used, and their cross-product (vertical, in purple) is shown varying as the angle between both vectors changes.
As you can see, when both vectors are separated by a right angle (90° = π/2 radians or τ/4 radians), their cross product’s vector reaches a maximum length, and when both vectors are parallel, their cross product is zero. (Here, both vectors a and b are unit vectors, so the magnitude of their cross product doesn’t grow beyond 1 either)
Similar to the way the dot product can be used to find if two vectors are perpendicular, one can use the cross product to find out if two vectors are parallel. You just have to check if these products are zero in each case.
In mathematical notation, we write |v| or ||v|| as the norm or magnitude of a vector v. With that notation, we can say that |a × b| = |a| |b| sin(θ), where θ is the angle from a to b.
The actual value of the magnitude |a × b| has a nice geometric interpretation: it is the area of the paralellogram made by the vectors a and b.
The cross product is said to be “anticommutative”, that is, the order used is important (so it is not commutative, in which case the order wouldn’t matter), and the “anti-” bit says that switching the order switches the sign of the product. Mathematically, a × b = -(b × a)
To be precise, the vector that results from the cross product is said to be a pseudovector, as it is not invariant through reflections: if you see your coordinate axes from a mirror, your right-handed coordinate system would become left-handed, so the definition of the vector breaks down. Vectors are extremely useful because they don’t depend on your system of coordinates, but pseudovectors do, and that’s why they are a caveat worthy of note. This is usually not a problem as long as you stick within the same handed-ness in different system of coordinates, by avoiding such reflections.
There are several ways to actually compute the value of the cross product between two vectors, but the most common one and easier to remember is by finding the determinant of a particular matrix.
Curiously enough, the definition of a vector product returning another vector is unique to 3 and 7 dimensions. More general but similar objects can be defined for other dimensions.
The cross product is very useful in physics for describing things such as torques (the rotational equivalent of a force), angular momentum (the rotational equivalent of linear momentum), and magnetic forces on charged particles (which act perpendicular to both the velocity of the particle and the magnetic field).
The physical nature of the vector quantity in such cases as torque or angular momentum can be tricky to understand without the proper insight, which is something that is rarely addressed by physics text books.
It is common for students to get stuck to the idea that a vector, as represented by the arrow, points to the direction the force or whatever it is acting or “going towards”. But for angular momentum and torque, this intuition breaks down.
The proper way to think about the vector for angular momentum is that the vector gives you an axis of rotation. The way the arrow is pointing tells you which of two possible ways the rotation is going (clockwise or counterclockwise, depending on your choice of coordinates and point of view). Using the right-hand rule (for a right-handed coordinate system), you can figure this out easily.
The magnitude of the vector is the actual magnitude of the momentum. So the vector is just a compact way to merge both bits of information on angular momentum in a single mathematical object. The consistency in all of the definitions is what makes it all work nicely, not coincidence.
The idea of a vector representing the axis and direction of rotation is the same here, but you can also, alternatively, consider it a plane in which the torque is acting on. The vector is normal to this plane.
But another tricky idea here is the dimensions of the torque vector. Remember that in physics, we say length, time and mass, for instance, are dimensions for a physical quantity. We say a meter, a second and a kilogram are units with the dimensions described before, respectively. This difference in terms (units vs. dimensions) is very important, and a lot of people don’t get it right the first time.
So, the dimensions of the torque vector are pretty weird: Newton-meter. A lot of students realize that this is the same thing as a Joule, which is a unit of energy. So why not say torque has the same dimension as energy, call it Joule, and get rid of the Newton-meter thing?
The answer is that while the dimensions match, the concepts don’t. Torque and energy are entirely different concepts, entirely different physical quantities, so they shouldn’t be treated the same even though their dimensions seem to match. But in my opinion, this difference is dogmatic if taken as Newton-meter vs. Joule, because it hides a very important detail.
I think torque makes more sense in units of Joules per radian. The radian is a dimensionless unit, which means it was hidden in there all along in our dimensional analysis. We were not comparing Joules with Joules, but Joules per radians with Joules! The radians bit comes from the fact torques act along an arc.
This is easy to see if we consider the work done by a torque τ: W = τθ, where θ is the angle the torque acted around, rotating an object. In this case, if you consider the dimensions of radians as non-disposable, you can easily see that it all works out.
The cross product is a very handy tool for defining some more complicated physical quantities. It may seem arbitrary at first, but the reasoning behind its definition is mathematically sound and extremely useful in practice.
In order to fully appreciate it, one must first get rid of a few intuitions on what vectors represent in physics. Vectors can represent a lot of things that are not explicitly directional, as you first start getting used to them, so the sooner you abandon that intuition the better.