On the direction of the cross product of vectors

One of my math professors always told me:

Understand the concept and not the definition

A lot of times I have fallen into this pitfall where I seem to
completely understand how to methodically do something without actually
comprehending what it means.

And only after several years after I first
encountered the notion of cross products did I actually understand what
they really meant. When I did, it was purely ecstatic!

Why on earth is the direction of cross product orthogonal ? Like seriously…

I mean this is one of the burning questions regarding the cross
product and yet for some reason, textbooks don’t get to the bottom of
this. One way to think about this is :

It is modeling a real life scenario!!

The scenario being :

image

When you try to twist a screw (clockwise screws being the convention)
inside a block in the clockwise direction like so, the nail moves down
and vice versa.

image

i.e When you move from the screw from u to v, then the direction of the cross product denotes the direction the screw will move..

That’s why the direction of the cross product is orthogonal. It’s really that simple!

Another perspective

Now that you get a physical feel for the direction of the cross product, there is another way of looking at the direction too:

Displacement is a vector. Velocity is a vector.
Acceleration is a vector. As you might expect, angular displacement,
angular velocity, and angular acceleration are all vectors, too.

But which way do they point ?

image

Let’s take a rolling tire. The velocity vector of every point in the
tire is pointed in every other direction. BUT every point on a rolling
tire has to have the same angular velocity – Magnitude and Direction.

How can we possibly assign a direction to the angular velocity ?

image

Well, the only way to ensure that the direction of the angular velocity
is the same for every point is to make the direction of the angular
velocity perpendicular to the plane of the tire.
Problem solved!