Write a vector as the combination of 2 other vectors

[MarcL]

Homework Statement

Write the vector u=(2,3,-1) as the sum of two vectors, one parallel to v=(1,0,-3) and the other orthogonal to v=(1,0,-3)

Homework Equations

orthogonal vector would imply that v"dot"w = 0 ( dot product)

Parallel vector would imply that v=kz
OR the crossproduct of both is 0

The Attempt at a Solution

My attempt went as follow:
I defined 2 vectors that w+z = u where u=(2,3,-1)

Vector w is orthogonal to v and therefore --> w"dot"v = 0 --> w₁*1+w₂*0+w₃*-3=0

And that z= kv which would give z= (1*k,0*k,3*k)

I thought that I could define any vector orthogonal that would give 0, for instance (3,1,1) would give 0.

However the system always ends up being inconsistent, I don't know where I might be going wrong :/

 

[jedishrfu]

For u = av + bw then w must be in the same plane as u and v and in addition it must be perpendicular to v as stated in your problem. The a and b are scalars.

Try drawing a picture and see if you can figure out w from the geometry and your vectors.

 

[MarcL]

two questions from this: is it obligatory to use a scalar in that equation for the w?

2) I don't see the point in understanding that w must be in the same plane as u and v, I mean I don't see how it helps me solve the problem.

Thing is, I chose an arbitrary vector for w that would satisfy the orthogonal requirement. However, using it, I get an a=-7 and b=3 which gives me a z parameter that is wrong :/

 

[jedishrfu]

There is a whole plane of vectors that are orthogonal to v but you must choose the one that is in the same plane as u and v right?

Try to visualize it and you'll see all three vectors must be in the same plane.

Another way to think of it is vector addition is a parallelogram with v and w as sides and with the diagonal between them being u the sum of the vectors.

 

[MarcL]

Is it kinda like spanning? I mean two vectors that add up to any vector in a plane?

 

[jedishrfu]

Yes that sounds right.

 

[MarcL]

but how would I go then to find my problem, for spanning I usually use a matrix to find my coefficient, but I don't know if that would apply in this case as it has to satisfy requirements, no?

 

[jedishrfu]

I'm thinking you should look at u - k.v = w and v.w=0

 

[Mark44]

For this problem you need to find the vector projection of u in the direction of v, which is denoted in some texts as ##\vec{Proj_v u}##. If you've been assigned this problem, it's a fairly safe bet that this concept has been presented in class. Once you have that vector, find a vector perpendicular to the vector projection, so that the two vectors you found add vectorially to u.

Draw a sketch. The two vectors you're looking for are the sides of a right triangle. Forget about matrices and spanning sets and focus on the geometry here.

 

[MarcL]

Well I can see where the projection comes from, how you use it geometrically. Only thing is that , in class, my teacher used a set of equation from the orthogonal and parallel requirement, so I assumed I had to do it that way, and that way I don't understand.

 

[jedishrfu]

You have u and you have the project of u on v so you have the lengths of two sides of a triangle with w being the third side.

 

[MarcL]

I realize I can do it with the projection, but we didn't use projection for that in class, so I believe my teacher would prefer if we solved it with system of equation(s). This is where I get confused, I don't know hjow to get it that way.

 

[jedishrfu]

I'd do it the way you know how and ask the teacher about the other method later. You might discover that that method doesn't even apply in this case.

 

[ehild]

Homework Statement

Write the vector u=(2,3,-1) as the sum of two vectors, one parallel to v=(1,0,-3) and the other orthogonal to v=(1,0,-3)

Homework Equations

orthogonal vector would imply that v"dot"w = 0 ( dot product)

Parallel vector would imply that v=kz
OR the crossproduct of both is 0

The Attempt at a Solution

My attempt went as follow:
I defined 2 vectors that w+z = u where u=(2,3,-1)

Vector w is orthogonal to v and therefore --> w"dot"v = 0 --> w₁*1+w₂*0+w₃*-3=0

And that z= kv which would give z= (1*k,0*k,3*k)

You wrote the equation u=w+z. Multiplying it with v ( dot product) you get ##\vec v \cdot \vec u = \vec v \cdot \vec w + \vec v \cdot \vec z ##. The first product is zero. z=kv. Find k.

 

[Mark44]

The problem works out pretty nicely if you organize your information. You're given vector u and v. The vectors you want are kv and w.

The three constraints yield a consistent set of equations so that you can find the scalar k and the coordinates of w.

1. kv + w = u
2. kv is perpendicular to w
3. Pythagorean theorem on the lengths of kv, w, and u

BTW, the method I mentioned earlier will also work, but as it hasn't been presented yet, probably shouldn't be used.

 

[haruspex]

The problem works out pretty nicely if you organize your information.

Hard to believe there's a simpler way than ehild's above.

 

[MarcL]

I just did this ( realized where my mistake was)

(2,3,-1)=a(1,0,3)+b(3,c,1) where the second vector comes from w1*1+)-3W3=0 --> W1=3W3 so I can just say 3,c,1,

Solve for 3 variables with 3 equation. I just didn't use the c but took any number at first, without understanding why ^^