Prove Vector Dependence: Coplanar \vec{u},\vec{v},\vec{w} Linearly Dependent

[spoc21]

Homework Statement

Prove that vectors $$\vec{u}$$, $$\vec{v}$$and $$\vec{w}$$ are coplanar if and only if vectors $$\vec{u}$$, $$\vec{v}$$and $$\vec{w}$$are linearly dependent

Homework Equations

The Attempt at a Solution

This may sound like an awkward question, but I am having much difficulty proving this. I tried to multiply each vector by a scalar, but got stuck, below is my solution, up until I got stuck.Any help is much appreciated!

We need to write $$\vec{u}$$= c₁ $$\vec{v}$$+ c₂ $$\vec{w}$$as:



[x₁, y₁, z₁] = c₁ [x₂, y₂, z₂] + c₂ [x₃,y₃, z₃]


[x₁, y₁, z₁] = [c₁ x₂, c₁ y₂, c₁ z₂] + [c₂ x₃, c₂ y₃, c₂ z₃]


x₁ = c₁ x₂ + c₂ x₃
y₁ = c₁ y₂ + c₂ y₃
z₁ = c₁ z₂ + c₂ z₃


This is where I got stuck, and I am not sure what to do next.

Thanks,

 

[gabbagabbahey]

Hmm.. I think the easiest way to prove this might be to realize that if 3 vectors are coplanar, then there will be some normal vector $\textbf{n}$ (the normal to the plane) that is perpendicular to all three and then look at some dot (and perhaps also cross) products.

 

[spoc21]

Thats basically the problem, I'm supposed to prove this without using either dot or cross products Also, do you think that my working is fine, or will not work?

Thank you very much

 

[Mark44]

This is an if and only if proof, which means you have to prove two things:
1. u, v, and w are coplanar ==> u, v, and w are linearly dependent.
2. u, v, and w are linearly dependent ==>u, v, and w are coplanar.

Maybe you already knew that.

To get you started, if u, v, and w are coplanar, then u = c₁v + c₂ w for some nonzero constants c₁ and c₂. So what can you say about the equation u - c₁v - c₂ w = 0?

 

[spoc21]

This is an if and only if proof, which means you have to prove two things:
1. u, v, and w are coplanar ==> u, v, and w are linearly dependent.
2. u, v, and w are linearly dependent ==>u, v, and w are coplanar.

Maybe you already knew that.

To get you started, if u, v, and w are coplanar, then u = c₁v + c₂ w for some nonzero constants c₁ and c₂. So what can you say about the equation u - c₁v - c₂ w = 0?

That means that they are linearly dependent...I completely understand the concept, but am stuck on the proof part; I'm a little confused on how to prove this using c₁ and c₂..

Thanks.

 

[gabbagabbahey]

To get you started, if u, v, and w are coplanar, then u = c₁v + c₂ w for some nonzero constants c₁ and c₂.

Not necessarily, if v and w are collinear, this need not be true. You need to examine two cases:

(1) At least two of the vectors are collinear
(2) None of the vectors are collinear.

 

[Mark44]

I was afraid that might not be enough. It's a real restriction not having access to the dot product and cross product.

 

[spoc21]

I was afraid that might not be enough. It's a real restriction not having access to the dot product and cross product.

I think you're on the right track, could you please elaborate on your method..

Thanks!

 

[Mark44]

Per gabbagabbahey's suggestion, look at two cases for u, v, and w.
1. At least two of the vectors are collinear.
2. None of the vectors is collinear.

In the first case, you can assume without loss of generality that u and v are collinear, i.e., that u = kv, for some constant k. Use that to show that the equation c₁u + c₂v + c₃w = 0 has a solution where not all the constants are zero.

In the second case, use what I said toward the end of post 4 to show that the vectors are linearly dependent (i.e., that the equation c₁u + c₂v + c₃w = 0 has a solution where not all the constants are zero).

 

[spoc21]

Per gabbagabbahey's suggestion, look at two cases for u, v, and w.
1. At least two of the vectors are collinear.
2. None of the vectors is collinear.

In the first case, you can assume without loss of generality that u and v are collinear, i.e., that u = kv, for some constant k. Use that to show that the equation c₁u + c₂v + c₃w = 0 has a solution where not all the constants are zero.

In the second case, use what I said toward the end of post 4 to show that the vectors are linearly dependent (i.e., that the equation c₁u + c₂v + c₃w = 0 has a solution where not all the constants are zero).

ok, so I get it up till c₁u + c₂v + c₃w = 0...How exactly do I show that c₁u + c₂v + c₃w is equal to 0, for non zero scalars c₁, and c₁..

I would really appreciate it if you could please help me with this,

thanks for your help

 

[Mark44]

Depends on which case you're working on. If it's the first case, u = kv, so replace u in the c₁u + c₂v + c₃w = 0 equation.

If it's the second case, see what I said in post 4.

 

[spoc21]

Depends on which case you're working on. If it's the first case, u = kv, so replace u in the c₁u + c₂v + c₃w = 0 equation.

If it's the second case, see what I said in post 4.

it is the second case, where u = c₁v + c₂w, for non zero scalars, I don't understand how to actually prove what you said in post 4..or writing that would be sufficient..

Thanks,

 

[spoc21]

ok, I get it...I was wondering how you got the u - c₂v - c₃w?

sorry, I just started vectors, and am a little confused with them..

 

[Mark44]

If u = c₁v + c₂w, then u - c₁v - c₂w = 0.

So here I have a solution to the equation c₁u + c₂v + c₃w = 0 where not all the constants are zero.

 

[spoc21]

If u = c₁v + c₂w, then u - c₁v - c₂w = 0.

So here I have a solution to the equation c₁u + c₂v + c₃w = 0 where not all the constants are zero.

ok, so I get 0 when I substitute u = c₁v + c₂w into the equation: is this correct?, since these combinations need to be linearly dependent
inorder to be coplanar, which we proved, as u = c₁v + c₂w...

 

[Mark44]

No, that's not it at all. Please tell me the definition of linear dependence, in the context of three vectors u, v, and w.

 

[spoc21]

No, that's not it at all. Please tell me the definition of linear dependence, in the context of three vectors u, v, and w.

Linear dependence:

c₁u + c₂v + c₃w= 0

 

[Mark44]

Haven't you noticed that exactly the same equation is used to define linear independence? IOW, there is more to dependence or independence than just that equation. What does the complete definition say?

 

[spoc21]

ok, so from what I've been taught, if you can express one of the vectors as a linear combination of the other vectors, then they are linearly dependent vectors...please remember that this is high school level math. I'm really confused on how to prove this, like I asked a tutor online , and he said that since these vectors are coplanar, so all we need to prove is that u - c₁v + c₂w = 0, so that c₁, and c₂ are non zero scalars...How could you prove this?
help is much appreciated..

Thanks!

 

[Mark44]

Linear dependence:

c₁u + c₂v + c₃w= 0

This is not how linear dependence is defined.

For example, consider u = <1, 0, 0>, v = <0, 1, 0> and w = <1, 1, 0>.

I can set c1u + c2v + c3w = 0, and that alone doesn't tell me anything, because this equation is always true if c1 = c2 = c3 = 0. For these vectors, however, it turns out that 1u + 1v - 1w = 0 is also true, so I have found a solution for the equation c1u + c2v + c3w = 0, where not all of the constants are zero.

Here's another example, this time with u and v the same, but w = <1, 0, 1>.

I can again set c1u + c2v + c3w = 0, which I can always do, no matter what vectors I'm talking about. If I solve for the constants this time, I have
c1<1, 0, 0> + c2<0, 1, 0> + c3<1, 0, 1> = <0, 0, 0>
==> c1 + c3 = 0
c2 = 0
c2 + c3 = 0

Substituting 0 for c2 in the 3rd equation gives me c3 = 0. Substituting 0 for c3 in the first equation gives me c1 = 0.

The only solution to the equation c1u + c2v + c3w = 0 is c1 = c2 = c3 = 0, and there are no other solutions.

The upshot of this is that for an equation c1u + c2v + c3w = 0, u, v, and w are linearly independent if the only solution is c1 = c2 = c3 = 0, AKA the trivial solution. For the same equation, u, v, and w are linearly dependent if there is some solutions with at least one of the constants not equal to zero. The solution with all three constants equal to 0 applies in both cases.

Now, if you have u = av + bw, is there a solution to the equation u1 - av - bw = 0 other than the trivial solution? There obviously is.

 

[spoc21]

Hi,

So this is what I am assuming (am very uncertain about it, and would appreciate it if you could rectify any mistakes):

lets assume that scalars: s and t, instead of c₁, and c₁ (to make it simple)

For these three vectors, u, v, andw to be coplanar:

u = sv + tw...(1)

For u, v, andw to be linearly dependent, we must have:

au + bv + cw (all scalars, not all of which are 0)rearranging the first equation (1), we get:

u = sv + tw

u-sv-tw = 0...(2)For constants we have 1(u), -s(v), and -t(w) = 0Since we have one constant which is not 0, 1..we can assume that vectors u, v, and w are linearly dependent..

So now we know that u, v, and w are only coplanar, when vectors u, v, and w are linearly dependent..

I really appreciate your help.
Thanks!

 

[Mark44]

Hi,

So this is what I am assuming (am very uncertain about it, and would appreciate it if you could rectify any mistakes):

This is mostly fine, but needs some tweaking.

lets assume that scalars: s and t, instead of c₁, and c₁ (to make it simple)

For these three vectors, u, v, andw to be coplanar:

u = sv + tw...(1)

For u, v, andw to be linearly dependent, we must have:

au + bv + cw (all scalars, not all of which are 0)

This is the part most in need of change. The above should be the equation au + bv + cw = 0, and there must be one and only one solutions; namely a = b = c = 0.

rearranging the first equation (1), we get:

u = sv + tw

u-sv-tw = 0...(2)

Yes.

For constants we have 1(u), -s(v), and -t(w) = 0

I don't know what you're trying to say here. Are you saying that 1u = 0? -sv = 0, -tw = 0? That's not the case at all.

Since we have one constant which is not 0, 1..we can assume that vectors u, v, and w are linearly dependent..

No, we don't assume that - we know that because of the definition of linear dependence.

So now we know that u, v, and w are only coplanar, when vectors u, v, and w are linearly dependent..

I really appreciate your help.
Thanks!

You have a lot of extra words that you don't need.

Starting with u = sv + tw...

u = sv + tw for some nonzero values of s and t
==> 1u - sv - tw = 0 has a non-trivial solution for the coefficients of the vectors; namely 1, -s, and -t
==> u, v, and w are linearly dependent.

 

[spoc21]

This is mostly fine, but needs some tweaking.
This is the part most in need of change. The above should be the equation au + bv + cw = 0, and there must be one and only one solutions; namely a = b = c = 0.
Yes.
I don't know what you're trying to say here. Are you saying that 1u = 0? -sv = 0, -tw = 0? That's not the case at all.
No, we don't assume that - we know that because of the definition of linear dependence.

You have a lot of extra words that you don't need.

Starting with u = sv + tw...

u = sv + tw for some nonzero values of s and t
==> 1u - sv - tw = 0 has a non-trivial solution for the coefficients of the vectors; namely 1, -s, and -t
==> u, v, and w are linearly dependent.

I'm trying to say that since we have 1 as a constant, we know that it is nonzero, confirming that the vectors are linearly dependent (am not sure, but was told by another expert)

Also, I was wondering if you could please elaborate on what you mean, when you say that the equation has only one solution, specifically, a=b=c=0...Could please elaborate on your reasoning a bit further, as I'm a little confused right now...

Help is much appreciated.
Thanks!

 

[Mark44]

OK, here's a simple example with the vectors u = <2, 1> and v = <-3, -1.5>. You should be able to realize that v = (-3/2)u or equivalently, that u = (-2/3)v.

v = (-3/2)u
<==> (3/2)u + 1v = 0
Since this is a linear combination of u and v that adds up to the zero vector and the coefficients of u and v are not all zero (in fact, neither one is zero), then by definition, u and v are linearly dependent. It's also worth noting that u and v are collinear.

Here's another simple example, this time with u = <2, 1> and v = <1, 2>. I set up the equation c1u + c2v = 0. Clearly c1 = c2 = 0 satisfies this equation, but I need to find out if there are other solutions (u and v lin. dependent) or this is the only solution (u and v lin. independent).

c1u + c2v = 0
<==> c1<2, 1> + c2<1, 2) = 0
<==> 2c1 + c2 = 0 and c1 + 2c2 = 0
By any of a number of methods, we can find that the only solution is c1 = 0, c2 = 0, so these vectors are linearly independent.

 

[spoc21]

Ok, I understand..thank you...I attempted the solution, and have attached it in pdf format, and would really appreciate it if you please take a look, possibly commenting on it (its not very long , just one page)..

help is much appreciated.
Thanks a lot!

 

[Mark44]

It is formatted very professionally, but the overall structure should be improved. I won't be able to look at this very closely for a few hours, but here are some initial thoughts.

1. This is an if and only if proof, so it should have two parts.
a. u, v, and w are coplanar ==> u, v, and w are linearly dependent.
b. u, v, and w are linearly dependent ==> u, v, and w are coplanar

2. In each of the two parts above, your first line should be the hypothesis (the part you are assuming is true - the part before ==>). Using logic and reasoning, you should get to the last line, which should be your conclusion (the part after ==>).

3. In 1a above, you need to examine two cases of coplanar vectors: one or more of the vectors are collinear; none of the vectors are collinear. You sort of did this, but there are some holes in your logic.

4. In your argument to show linear dependence you have
u = sv + tw
Then you say "Slightly rearranging equation (1), ..." followed by two more lines.

Instead of saying what you're going to do (since it should be obvious), just do it.
u = sv + tw
<==> u - sv - tw = 0
These two equations are obviously equivalent since all I did was subtract (sv + tw) from both sides. Don't belabor baby steps. Your instructor will get it.

5. The last equation above shows that u, v, and w are linearly dependent, by definition. Any time the equation au + bv + cw = 0 has a nontrivial solution, the vectors are linearly dependent. In this case, a = 1, b = -s, and c = -t. By nontrivial, I mean any set of constants that are not all zero.

6. You have three sections that start "Proof:". There should be only two of these - one for the statement in 1a and one for the statement in 1b.
At the very top you should state the thing you are going to prove.
Vectors u, v, and w are coplanar if and only if they are linearly dependent.
Under that you should have the "only if" part:
If u, v, and w are linearly dependent, then they are coplanar.
Proof: ...

Then the "if" part:
If u, v, and w are coplanar, they are linearly dependent.
Proof: ...

This is the usual ordering of these two parts, but as long as you prove both, the ordering doesn't matter.

 

[spoc21]

ok, thanks...basically, my proof is correct, there are just a few formatting errors right..I will fix these things right away...but apart from that, there aren't any solution errors on my part..correct? Again, thank you..

 

[Mark44]

It's more than just formatting errors - there are some problems with the structure, as noted above.

 

[spoc21]

thanks!