Dot product and cross product Intuition

Question

I know that the dot product and cross products of two vectors represent their components acting in each other's direction but it always puzzles me to think why and how, particularly what bangs my head the most is the intuition for cross product. Please answer in an intuitive manner for both along with a few examples of real life situations. Any answer will be of great help. If someone refers to the images given on Wikipedia articles for the products, what I want is an intuitive explanation of geometric references.

Answer 1

Your intuition that the dot product is the component of one vector along the other is good. Continue with that. Specifically, you probably know that you can break up a vector as a sum of components, each of which points along a coordinate axis, yes? Each of those components can be written as some scalar number times the unit vector of the respective axis, yes? The dot product of a vector with that unit vector tells you the value of that scalar number; in a sense, "how much" does the vector point in that direction. Well, there's nothing in the geometry of space that picks out any particular axes as special, so try to imagine breaking up one of the two vectors in your dot product into a component which is parallel to the other and another component which is perpendicular to the other. The dot product then tells you the number which expresses how much one vector lies in the direction of the other, scaled up (that is, multiplied) by the magnitude of the other vector.

Now, cross products are a totally different thing. You are not combining two vectors to get a number that tells you something about the magnitude of their overlap. You are instead combining two vectors and getting a new third vector which is perpendicular to the other two, with a magnitude that tells you something about the magnitude to which they are perpendicular. At least that's how it's usually taught. I've always found that confusing, but it is standard.

Another view of the cross product is that it's telling you about the plane that the two vectors lie in; specifically how much one vector "circulates in the plane" towards the other. By analogy: a vector is a one-dimensional object, pointing along a line in space, with an amplitude telling you how much it points that way; a number is like a zero-dimensional point; and there are another types of objects that represent two-dimensional areas and three-dimensional volumes. However, it turns out that since the two-dimensional areas have a direction (one dimensional!) which is perpendicular to them, they can be described by something that behaves like a vector. (And three-dimensional volume elements are really just a magnitude, so they can be described just by a number, like a point.) The type of object that a cross product really is, is the two-dimensional analog of the one-dimensional vector. (They are called 2-forms, but that's not important. In this language, a vector is a 1-form, and a number is a 0-form.) It's a little planar object, not a little pointy object. However, since we can introduce students to the full range of zero- through three-dimensional objects by using just numbers and vectors while avoiding these more complicated structures, that's what teachers usually do. However, I find that understanding that the cross product is giving me a little circulating two-dimensional bit in the plane defined by the two vectors helps my understanding more than pretending that the cross product is a vector.

That said, some of what I said about the cross product is very special to three dimensions. In two dimensions, the cross product still gives you a 2d object, but since your whole space is two-dimensional, that 2d object is like a little volume element, which can be expressed as just a number, a magnitude. So, in two dimensions, it's standard for your teacher to tell you that the cross product is just a number, like the dot product is. So, it's a number in 2d and a vector in 3d. OK, fine, but really it's a 2-form in both.

It's worth noting that if you work in geometries that are larger than 3 dimensions, the dot product is always a number that tells you about overlap, but the entire notion of the cross product as a vector is out the window; doesn't work. You need to think of the cross product (usually called a "wedge product" or "outer product" in this context) as a two-dimensional object. (And, yes, in these higher dimensions, there are additional product type that take p vectors and give you an p-dimensional output object, an "p-form", as long as p is less than or equal to the dimension n of the space. And, usually, p-forms in n-dimensional space are "dual" to (n-p)-forms; for example, 2-forms in 3-space are dual to 1-forms in 3-space.)