How Can I Prove Vector Identities Using Algebraic Manipulation?

[doggbAT]

Homework Statement

Question One: Prove that |u x v|^2 = (u . u)(v . v)-(u . v)^2 where u and v are vectors.

Question Two: Given that u = sv + tw, prove algebraically that u . v x w = 0 where u, v and w are vectors and s and t are integers.

Homework Equations

I don't know :(

The Attempt at a Solution

I have expanded the equations, expressing vectors as [x,y,z].. I have no direction after that. I am curious how you would explain that the cross product of two vectors = 0 without the vectors having any value. Any help is greatly appreciated.

 

[AEM]

Homework Statement

Question One: Prove that |u x v|^2 = (u . u)(v . v)-(u . v)^2 where u and v are vectors.

Question Two: Given that u = sv + tw, prove algebraically that u . v x w = 0 where u, v and w are vectors and s and t are integers.

Homework Equations

I don't know :(

The Attempt at a Solution

I have expanded the equations, expressing vectors as [x,y,z].. I have no direction after that. I am curious how you would explain that the cross product of two vectors = 0 without the vectors having any value. Any help is greatly appreciated.

I think that expressing your vectors in terms of x, y, and z is the wrong way to go about it. Try recalling the definition of the magnitude of the cross product as the magnitude of each of the vectors times the sine of the angle between them. (Note that you're squaring the magnitude of the cross product.) Now u dot u is just u squared, etc and what is another way to write sine squared? (Hint: it uses a familiar trig identity). This should get you started.

 

[doggbAT]

Thank you :D I've answered the first question, and am now working on the second question. Your help worked perfectly!