does anyone know how to solve this problem?

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dgwlnhkbw

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I have no idea about this proof. I tried the scalar/vector projection equation to find something helpful but I dont know how to star.

Let u and v be two non-zero vectors in R2 that are not parallel to each other. Let a, b be two real numbers. Prove that there is exactly one vector r in R2
such that its scalar projection on u is a and its scalar projection on v is b.
 
dgwlnhkbw said:
I have no idea about this proof. I tried the scalar/vector projection equation to find something helpful but I dont know how to star.

Let u and v be two non-zero vectors in R2 that are not parallel to each other. Let a, b be two real numbers. Prove that there is exactly one vector r in R2 such that its scalar projection on u is a and its scalar projection on v is b.
Click to expand...
You might start by proving existence, by constructing a vector r that fits the requirements.

But we can help a lot more once we see what you have already done, and can either correct it or suggest a next step.
 
Subhotosh Khan said:
You said: Please share your work - so that we know where to begin to help you.
Click to expand...
IMG_E2DACB394129-1.jpeg

I think the r vector should locate between u and v vectors. The angle between u and v should equal to the angle between u and r plus the angle between v and r. If I can get r or |r| equal to something consisted of those existing vectors and numbers (u, v; a, b), then it should have an r vector. I do get something as I expected but it was just my assumption. I don't know whether it works.
 
Dr.Peterson said:
You might start by proving existence, by constructing a vector r that fits the requirements.

But we can help a lot more once we see what you have already done, and can either correct it or suggest a next step.
Click to expand...
see above! Thanks!!
 
In general, r will not be between u and v, depending on what a and b are; here is an example:

1600735725914.png

Here u_a and v_b are the vector projections of the desired r on u and v; that is, they are vectors in the direction of u and v, respectively, with lengths a and b. You can see how I obtained r, which has those vector projections. To do this, just think about what a scalar projection and a vector projection are, as in the picture. (Your picture is fine, and there's nothing wrong with doing your initial thinking in the simplest case, so you don't need to use the specifics of my picture to do the work. But I wouldn't use the angles, which may not be where you think they are.)

In your work, there is some complete nonsense, particularly your equation for r. How do you think you can divide a scalar by a vector?? You're trying to do ordinary algebra with vectors, which you just can't do.
 
You are given that u and v are not parallel so one is not a multiple of the other. That means that they are independent and so from a basis for \(\displaystyle R^2\). Any two vectors in \(\displaystyle R^2\) can be written as a linear combination of u and v!
 
HallsofIvy said:
You are given that u and v are not parallel so one is not a multiple of the other. That means that they are independent and so from a basis for \(\displaystyle R^2\). Any two vectors in \(\displaystyle R^2\) can be written as a linear combination of u and v!
Click to expand...
you mean something like au+bv=r? someone told me that this question can be transferred to this equation, but Idk why..
 
One thing that will help us would be to know what sort of things you have learned about vectors. Are you in a linear algebra class, or have you focused mostly on geometric/trigonometric aspects, or something else?

I don't immediately see a simple connection between your problem and taking u and v as a basis, but that may be because I'm thinking more geometrically. On the other hand, the fact that you are to prove existence and uniqueness, without necessarily having to find a formula for r, could fit with using whatever theorems you have learned, and may be considerably easier. What theorems do you have?
 
Dr.Peterson said:
One thing that will help us would be to know what sort of things you have learned about vectors. Are you in a linear algebra class, or have you focused mostly on geometric/trigonometric aspects, or something else?

I don't immediately see a simple connection between your problem and taking u and v as a basis, but that may be because I'm thinking more geometrically. On the other hand, the fact that you are to prove existence and uniqueness, without necessarily having to find a formula for r, could fit with using whatever theorems you have learned and maybe considerably easier. What theorems do you have?
Click to expand...
I am in an advanced calculus I class and just finished vector, dot product and projections.
 
Okay, that implies that the idea of a basis is not emphasized, and what theorems you have probably deal with the geometric aspects.

Have you tried applying the ideas implied in my answer? Think about my dotted lines, and how they can be used to find r. (You don't necessarily have to find a formula for r, just to convince yourself that will always be one such r.)
 
Given some vector, r, how would you find the "projection of r on u" and how would you find the "projection of r on v"? Write down the formula for each, set the first equal to "a" and the second equal to "b". Solve the two equations for the components of r. If the answers are unique, you are done!
 
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