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Proof: Relationship between cross product and sin of angle

Proof: Relationship between the cross product and sin of angle between vectors. Created by Sal Khan.

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• Wait, so we have:

||a x b|| = ||a|| . ||b|| . sin(theta)

Isn't that a NUMBER? The result of that calculation - a length, times a length, times a sine should be a number.

In the last video Sai said that the cross product of two vectors is STILL a vector. It doesn't look like a vector to me.

Is it still a vector?
• Wait, I figured it out.

||a x b|| is THE LENGTH of (a x b), not the vector itself. That is a number.
• For the basic equation |AxB|=|A||B|sin(theta), what if sin(theta) is negative? The lenght of a vector is a positive scalar by definition, so if sin(theta) is negative, what does it mean?
• Good question! Recall that the angle, x, between two vectors can only be between 0 and 180 degrees, inclusive. When x is between 0 and 180, sin(x) is between 0 and 1 (think of the unit circle). Therefore, the sine of the angle between the two vectors will never become negative number, and therefore |AxB| will always be positive, as required.
• At Sal begins to factor out the squared "a" terms of the equation and at Sal begins to take the square of the dot product identity--but how do we know to do this? When my professors pull these tricks out, it's amazing--but how did they know to do them? Obviously, they learned them from someone else, but the people who first "discovered" these relationships--how did they know to perfectly combine these equations to come up with such an elegant result? That level of intuition escapes me.
• They probably fooled around with it in any way possible to make it equal. It is hard work not magical thinking.
• Is there any difference between llall and lal? Or do they both refer to the magnitude?
• Akib,

The way most people use the notation, ||a|| typically refers to magnitude of a vector, and |a| refers to the absolute value of a scalar. Since both vectors and scalars are usually denoted by lower case letters, it's important to understand which one represents which in the book you are using or the test you are doing.

By the way, not every book or test uses exactly the same notation. Some books might have a special symbol over the letter to make it clear that it is referring to a vector. Other books might use bold print for vectors and matrices and regular print for scalars. Other books might use only greek letters for scalars, and latin letters for vectors and matrices. So if there is a question about what represents what, it is important to ask.
• While I can understand where this proof is coming from, I always ask myself how anyone found this proof. Is there any sort of intuition you can develop for proof-building?
• The only way to get better at proof-writing is practice. Basically every proof-writing problem is unique, but there are broad techniques that you can learn that will usually give you a good starting point.

https://math.berkeley.edu/~hutching/teach/proofs.pdf
• Can someone provide link to the video where Khan explains intuition behind |a x b| etc? ( onwards)
• at , Sal says that R3 is the only place where the cross product is defined. I understand that it is not defined in R2, but what about R4?
• Sal was simplifying a little bit -- the cross product is also defined in 0, 1 and 7 dimensions (although in 0 and 1 dimensions it is not at all useful because any cross product is always equal to the zero vector). So you could have a cross product in R^7. However, it doesn't work in R^2, R^4 or any other dimension. The reason for this is extremely complicated, but essentially it boils down to the fact that division only works in 1, 2, 4 and 8 dimensions. You already know 1-dimensional division -- it's just ordinary division. You probably know about 2-dimensional numbers (imaginary numbers), and then 4- and 8-dimensional numbers are called quaternions and octonions, and work in a similar way to the imaginary numbers.