Math Is Fun Forum

  Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

#1 2007-06-26 09:20:30

dukebdx12
Member
Registered: 2007-03-21
Posts: 9

Proving a vector property

I am in calculus 3 and the chapter on vectors (cross product) one of the properties is:
u x v = 0 if and only if u + v are scalar multiples of each other

I need to prove u x v = 0

A hint we have is:
  u x v    =    0
assume -> deriv ; and then take it back the other way.

Offline

#2 2007-06-27 14:25:53

George,Y
Member
Registered: 2006-03-12
Posts: 1,379

Re: Proving a vector property

are u and v both 3 dimensional?


X'(y-Xβ)=0

Offline

#3 2007-06-27 14:30:32

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Proving a vector property

As I believe George was hinting to, write out the vectors piecewise, do all the necessary operations, and the answer should appear.

For example, going the one way would be:

let u = <x, y, z>
v = <ax, ay, az>

Now compute the cross product.

The other way, start with two vectors:

u = <x, y, z>
v = <a, b, c>

And compute the cross product again, and set it to 0.  Show that this resulting vectors shows that you have write u as a scalar multiple of v.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Offline

#4 2007-06-27 18:44:12

luca-deltodesco
Member
Registered: 2006-05-05
Posts: 1,470

Re: Proving a vector property

alternatively, rather than doing it piecewise, you could just write:

this will be 0 when sinθ = 0, which is when the two vectors a and b are collinear. if they are collinear then they can only be multiples of eachover. The only other time the two vectors can give a 0 vector, is if one or both of them is a 0 vector, in which case they need not be collinear, but then the other is still a multiple of the other by 0.

Last edited by luca-deltodesco (2007-06-27 18:48:18)


The Beginning Of All Things To End.
The End Of All Things To Come.

Offline

Board footer

Powered by FluxBB