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# Cross product introduction

Introduction to the cross product. Created by Sal Khan.

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• At There is no proof for that definition?
• Definitions don't get proofs...

A theorem says: 'IF this thing is true THEN that thing must also be true'. Theorems require proof.

A definition says: "IF you have stuff that looks like this, THEN you may call it that'. Definitions don't require proof... they are either used properly or improperly and they are either useful or not, but they don't need to be proved.

Example: Definition... IF a polygon is a triangle and one of it's angles is exactly 90 degrees, THEN that triangle may be called a right triangle.

Definitions just give you smaller names for a collection of facts... if you say a figure is a right triangle then you are saying that figure also embodies all of the characteristics of a right triangle. If you are wrong then the figure is not actually a right triangle, but that does not 'disprove' the definition.

However... THEOREM: If a triangle has two angles that sum to exactly 90 degrees, then that triangle is a right triangle. This is a theorem and would have to be proven. And if you could find a triangle with two angles that sum exactly to 90 degrees and is not a right triangle, then you would have disproved this theorem in all cases and for all time (not that I'm particularly worried about it).

A definition renames information.

Cheers.
• Here I learn that sal is both a great teacher and a pretty good artist
• I wish this course had some exercises.
• What happens in cross multiplication with more than 3 rows? For example 4 rows, 5 rows, and etc?
• In some cases it is easier to think of the result of something to tell you why it's undefined. In R^3 we get 2 vectors from the cross product (both which are orthogonal to the original vectors (one is just the negative of our output)).
However in R^4 or greater. there are multiple orthogonal vectors, as operators only return one value, it is impossible to define a singular answer (and by extension any answer) to a cross product in R^4 or greater.
• Could someone please explain to me the point of the Cross Product? Why do we have it and what are some examples of the Cross Product in action?

I'm learning Linear Algebra because of its importance in Economics and Statistics and none of my textbooks even mention the Cross Product--all I know is that it has something to do with Physics and I'm curious as to what that something is... Thanks!
• Steve,

Here is a great example of the cross product in action. Imagine that you have a door and you want to apply a force to the door to open it. Now you could apply that force anywhere on the door… maybe you apply it near the side of the door that has the hinge, or maybe you apply the force far away from the hinge. Also, you could apply the force in any direction-- maybe you apply it perpendicular to the plane of the door, or maybe you apply it parallel to the plane of the door.

What happens? Well, if you apply the force parallel to the door, the door doesn't swing open. Instead, it's like you are pushing or pulling the door towards or away from the door jamb and the hinge. Kind of like you are trying to pull the door off the hinges.

Suppose instead you apply the force perpendicular to the door… but on the side of the door near the hinge? Well now the door will swing open… but it takes an awful lot of force to move the door. It would take much less force to open the door if you pushed perpendicularly on the door on the side of the door far away from the hinge.

What moves the door is something called a 'moment,' and moment is the cross product of a Force vector and a distance or position vector. In other words, not only does the force need to be at some distance from the point about which we are creating the moment, but it also needs to be applied perpendicularly to that distance.

It turns out that the cross product describes this relationship perfectly. The cross product of two parallel vectors is 0, and the magnitude of the cross product of two vectors is at its maximum when the two vectors are perpendicular.

There are lots of other examples in physics, though. Electricity and magnetism relate to each other via the cross product as well. For example, if you move a wire through a magnetic field, it will produce a current in the wire that moves perpendicularly to the magnetic field.
• Bro just became picasso for a moment then moved on
• how does multiplying a×b×a and a×b×b and getting 0 out of it prove me that a×b is orthogonal
• I think you are getting a little confused about similar terms which is really easy to do so I'm going to state some definitions first.

a and b are both vectors, the video talks about two different operations you can do on vectors, Cross Product (which it introduces and the Dot Product which it expects you to already know.

The Dot Product of two vectors gives a scaler, let's say we have vectors x and y, x (dot) y could be 3, or 5 or -100. if x and y are orthogonal (visually you can think of this as perpendicular) then x dot y is 0. (And if x dot y is 0 x and y are orthogonal).

The Cross Product, the new one in this video, of two vectors gives a new vector not a scaler like the dot product. So if we say x and y are vectors again then x cross y = z and z is a vector of the same size as x and y. It's a special vector, though, because it is orthogonal to x and y. This isn't magic, the cross product is defined to cause this result. (The video mentions you might notice a relationship between determinants and how it is defined, but you aren't expected to understand the construction of the definition of cross product at this point)

So the video has vectors A and B and it creates AxB. This new vector AxB is orthogonal to A and it is orthogonal to B because that's what the cross product does. That means AxB (dot) A =0 and AxB(dot) B=0. The video then does the calculations to show that both of those statements are true. Since AxB (dot) A does turn out to equal 0 and AxB (dot) B =0, we have confirmed that AxB, the vector created with the cross product, is in fact orthogonal to A and orthogonal to B.

(orthogonal is a relationship between two vectors, not a property of a vector, so axb can't be orthogonal, but it can be orthogonal to a. Like a line in the x y plane can't be perpendicular, but it could be perpendicular to another line)

Hope this helps. If any of it isn't clear, reply and I can try to say it another way.
• Does this imply that vector a cross vector b is different to vector b cross vector a?
• Yes, they are different, but share a simple relation. a x b = negative b x a.
• How is linear algebra used in economics?
• You can Google "linear algebra in economics" to get a more complete picture, but it's hard to understate the importance. For a single elementary example, here is a video using the Leontief input-output model to solve a problem.
https://youtu.be/-1jT5NOk93w
• How does one find a vector perpendicular to two linearly independent R5 vectors?