Expressing Vector v in the Same Plane as u1 and u2

[Srumix]

Homework Statement

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

Homework 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?

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.

I got the answer

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

Thanks in advance!

 

[Dick]

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

 

[amaresh92]

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

 

[HallsofIvy]

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

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.

 

[Srumix]

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

Thanks a lot for your replies!