# Determine if three vectors lies in the same plane. (Linear Algebra - Introduction)

1. Dec 31, 2009

### Srumix

1. The problem statement, all variables and given/known data

The problem is to decide that the vector v = (2, -7, 1) lies in the same plane M as the vectors u1 = (2, -1, 3) and u2 = (1, 1 ,2) and express v as coordinates in the base u1, u2

2. Relevant equations

I decided to utilize that if the three vetors lies in the same plane, they must be linear dependent. If we assume that u1 and u2 lies in the same plane as v then we can express v as:

v = s1u1 + s2u2

Then

s1u1 + s2u2 - s3v = 0

If they are linearily dependent they must lie in the same plane. Right?

3. The attempt at a solution

I solved the linear equation that the above relationship gives and finds that one of the equations is 0 = 0, hence it has infinitely many solutions, i.e s1 and s2 can be arbitraily chosen. (Right?)

I could now write v as coordinates in u1 and u2

By solving

s1u1 + s2u2 = (2, -7, 1)​

for s1 and s2.

v = 3u1 - 4u2

Which was correct, according to the book.

Now, i would like to know if this is a correct way to solve this problem, or if i was just getting lucky. This is my first attempt att liear algebra so I have no good intuition about the methods to solve this types of problems.

Please be as critical as you can!

Please point out any faults or "fuzzy thinking" I've done :)

Happy New Years!!

P.s

2. Dec 31, 2009

### Dick

Re: Determine if three vectors lies in the same plane. (Linear Algebra - Introduction

That seems like a perfectly fine solution to me. You seem to be pretty clear on all of the steps.

3. Dec 31, 2009

### amaresh92

Re: Determine if three vectors lies in the same plane. (Linear Algebra - Introduction

I think it will be better to use scalar triple product for coplanarity .

4. Jan 1, 2010

### HallsofIvy

Staff Emeritus
Re: Determine if three vectors lies in the same plane. (Linear Algebra - Introduction

I disagree.

1) This is linear algebra and the scalar triple product is only defined for R3.

2) The problem also asked that v be written in terms of u1 and u2 and the scalar triple product doesn't help you do that.

5. Jan 1, 2010

### Srumix

Re: Determine if three vectors lies in the same plane. (Linear Algebra - Introduction

Good to know that my method to solve this problem was correct.