Finding Constants for Linear Dependence in 3D Vectors

[bryanosaurus]

Going through a mathematical physics book in the section about vector spaces, in the section showing how to prove vectors are linearly dependent their example is:

Two vectors in 3-d space:

A = i + 2j -1.5k
B = i + j - 2k
C = i - j - 3k

are linearly dependent as we can write down

2A - 3B + C = 0

I understand the concept of linear dependence, and why the answer makes sense (non-zero constants exist) but my question is how they determined the constants needed to show the vectors are dependent. My first thought was Gaussian elimination but I don't think that's correct.

Any help would be appreciated.

 

[Vid]

You're right in suspecting Gaussian elimination as one way to find them, but can you figure out why?
Start with xA + yB + zC = 0 where x, y, and z are the unknown constants and try and solve for them. You should find a system of three equations in three unknowns.

Also, welcome to PF.

 

[bryanosaurus]

Thank you. So I get something like this (eliminating x):

x + 2y - 1.5z = 0
-y - .5z = 0
-3y - 1.5z = 0

I can't remember how to solve a set of equations like this where they are all set to zero.
I thought the process was once a variable is eliminated, to solve for say cy = z
then set z=t and try to plug back to find x. When I do this, I do not come up with 2, -3, 1 or any multiples of them.

 

[Vid]

Your starting equations should have been:
x + y + z = 0
2x + y - z = 0
-1.5x - 2y - 3z = 0

From this, put it into a matrix and use Gaussian elimination. If you don't know what I'm talking about, you should start from the beginning of linear alg.
Here's some notes for a introduction to linear algebra class for reference.
http://tutorial.math.lamar.edu/Classes/LinAlg/LinAlg.aspx

 

[bryanosaurus]

Your starting equations should have been:
x + y + z = 0
2x + y - z = 0
-1.5x - 2y - 3z = 0

I know how to use Gaussian, but when I originally worked it out I had put the starting equations in wrong. That's why in the OP I thought I was wrong for using that method. Now that you posted the correct starting equations, I see my error. Thanks a lot, I sure wasted a lot of time getting hung up on a simple problem haha.