Proof this cross product property

[-EquinoX-]

Homework Statement

Show that :

|| a x b||^2 = ||a||^2||b||^2 - (a . b)^2

Homework Equations

The Attempt at a Solution

I don't know where to start off

 

[Dick]

What's the difference between ||axb||^2 and (axb)^2? Don't you mean (a.b)^2 instead of (axb)^2? Just use expressions for the magnitude of the cross and dot product and the angle between the vectors a and b.

 

[Mark44]

What you're supposed to show doesn't make sense.
From your title I assume you're asking about the vector cross product for vectors in R^3.
Given that, I understand what is meant by || a x b||^2, ||a||^2, and ||b||^2, but I have no idea what (a x b)^2 even means, let alone how to compute it.

Can you clarify what is meant here by (a x b)^2?

 

[-EquinoX-]

sorry a typo, I've revised the question, it should be (a.b)^2

I know that a x b = ||a|| ||b|| sin x, so then how do I find the magnitude of this? I am confused because there's the sin function...

 

[Dick]

Then express both side in terms of |a|, |b| and the angle between them.

 

[-EquinoX-]

not sure I understand what you mean

 

[Dick]

If t is the angle between a and b, then a.b=|a|*|b|*cos(t). There a similar relation for |axb|.

 

[-EquinoX-]

ok is this correct?

$$|a\times b|^2 = |a|^2|b|^2\sin^2\theta = |a|^2|b|^2(1 - \cos^2\theta)$$
$$= |a|^2|b|^2 - |a|^2|b|^2\cos^2\theta = |a|^2|b|^2 - (a\cdot b)^2$$

 

[Dick]

ok is this correct?

$$|a\times b|^2 = |a|^2|b|^2\sin^2\theta = |a|^2|b|^2(1 - \cos^2\theta)$$
$$= |a|^2|b|^2 - |a|^2|b|^2\cos^2\theta = |a|^2|b|^2 - (a\cdot b)^2$$

Absolutely.

 

[-EquinoX-]

I am not sure of the notations through can I write |a x b|^2 as ||a x b||^2 as it denotes magnitude?

 

[Dick]

They both mean the same thing to me. If you're a stickler you might want to demand writing ||v|| if v is a vector and |x| if x is a number.

 

[-EquinoX-]

well yes, but what's meant here by ||v|| here is the magnitude of vector v right? now I am confused though, by definition of the geometric, it says that:

a x b = ||a|| ||b|| sin x

and it's not

|a x b| = ||a|| ||b|| sin x

can you explain this?

 

[Mark44]

well yes, but what's meant here by ||v|| here is the magnitude of vector v right? now I am confused though, by definition of the geometric, it says that:

a x b = ||a|| ||b|| sin x

and it's not

|a x b| = ||a|| ||b|| sin x

can you explain this?

I don't understand what you're saying above. Does "and it's not" refer to the first equation or the second? Either way, the first equation is incorrect and the second one is correct (aside from inconsistent usage of | | and || ||).

a X b is a vector, which ||a|| ||b|| sin x is the product of three numbers, and so is a number.

 

[natangwe]

Can you please prove for me this identity: A x (B x C) = (A.C)B - (A.B)C

 

[natangwe]

Can you please prove for me this identity, analytically: A x (B x C) = (A.C)B - (A.B)C and llU x Vll^2 = llUll^2 llVll^2 - (U.V)^2. Any one with understanding on this please help out.