Two impossible crossproduct problems

[Nikitin]

Hi, I really need some help here.. Vectors remind me why I hate geomtery.

problem 1: Prove that |UxV|² = |U|²+|V|² - (U*V)²

How can I prove that these two are equal without spending 1 hour using algebra? Maybe there is some geometry quirk that I'm not seeing?

problem 2: We have two vectors A=[1,1,1] and B=[1,2,3]

Find a vector C so that AxB = AxC, where C =! B.

I tried using algebra on this but I just ended up with crazy expressions for C_x, C_y and C_z where each of them were dependent on the others.

So.. Is there some other way? All help is appreciated =)

 

[Nikitin]

I would also like to point out that I haven't learned yet about things like the ratio between sine(x) and cosine(x), so pls if possible tell me this can be solved by simple algebra or by using geometry?

 

[Mute]

For the second one, since it just asks for a vector C not equal to B, the easiest thing to do would be to choose C = B + V, where V is some vector such that A x V = 0. Do you know what kinds of vectors have that property?

 

[Fredrik]

problem 1: Prove that |UxV|² = |U|²+|V|² - (U*V)²

How can I prove that these two are equal without spending 1 hour using algebra? Maybe there is some geometry quirk that I'm not seeing?

Do you know how to rewrite U·V and |U×V| in terms of |U|, |V| and the angle between U and V. You need to use those identities, and also $\cos^2x+\sin^2x=1$.

 

[Nikitin]

For the second one, since it just asks for a vector C not equal to B, the easiest thing to do would be to choose C = B + V, where V is some vector such that A x V = 0. Do you know what kinds of vectors have that property?

ahh, so AxC = Ax(B + V) = AxB + AxV where AxV=0.

Very cunning. Yes, V equals any vector which is pararell with A. Thank you 4 the help!

Do you know how to rewrite U·V and |U×V| in terms of |U|, |V| and the angle between U and V. You need to use those identities, and also $\cos^2x+\sin^2x=1$.

Sure, |U|²*|V|² - |U|²*|V|²*cos(x)²= |U|²*|V|²(1-cos(x)²)=|U|²*|V|²(sine(x)²)=|UxV|²

correct? tho we haven't learned about the 1=cos(x)^2 + sine(x)^2 trick in my maths class so maybe there is another way to prove it?

thank u very much 4 the help anyways