Determining 3D Vector Basis with a,b,c Vectors

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Homework Help Overview

The discussion revolves around determining if the vectors a = (2, -3, 2), b = (1, 1, -1), and c = (8, 5, -2) can form a basis for R^3. Participants explore concepts related to vector independence and the properties of various vector operations.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss using the dot product, cross product, and scalar triple product to assess the vectors' properties. Questions about the definition of a basis and the necessity of orthogonality for linear independence are raised.

Discussion Status

There is an ongoing exploration of the properties of the vectors, with some participants suggesting specific mathematical operations to investigate their independence. Multiple interpretations of the problem are being discussed, particularly regarding the definitions and requirements for a basis.

Contextual Notes

Some participants express uncertainty about the definitions and properties involved, indicating a need for clarification on the concept of a basis and linear independence. There is also mention of the original poster's confusion regarding orthogonality.

kevykevy
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Homework Statement


Determine whether the the vectors a = (2, -3,2), b = (1, 1, -1) and
c = (8, 5, -2) can be used as a basis for vectors in R^3 (3D space)


Homework Equations





The Attempt at a Solution


I really have no clue, I think maybe you use either cross product, dot product or triple scalar product...?
 
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kevykevy said:
I think maybe you use either cross product, dot product or triple scalar product...?

Why don't you try one of these? I'd use the dot product first, to show whether or not the vectors are mutually orthogonal.
 
kevykevy said:

Homework Statement


Determine whether the the vectors a = (2, -3,2), b = (1, 1, -1) and
c = (8, 5, -2) can be used as a basis for vectors in R^3 (3D space)

What's the definition of a basis?
 
cristo said:
Why don't you try one of these? I'd use the dot product first, to show whether or not the vectors are mutually orthogonal.

They don't have to be mutually orthogonal to be linearly independent, and it is unlikely that they will be. To kevykevy: You were on the right track with the scalar triple product. What properties of this product do you know?
 
LeonhardEuler said:
They don't have to be mutually orthogonal to be linearly independent, and it is unlikely that they will be. To kevykevy: You were on the right track with the scalar triple product. What properties of this product do you know?

Sorry, I read "orthogonal" that wasn't in the question!
 
cristo said:
Sorry, I read "orthogonal" that wasn't in the question!

I know what you're talking about. I've been there more than a few times myself. :redface:
 
Cross Product
a x b = (1, 4, 5)

Dot Product
(1, 4, 5) x (8, 5, -2) = 18

Since 18 doesn't equal 0, then the vectors cannot be used as basis vectors

is that right?
 
to radou - basis vectors, example i, j, and k with the carot(^) on top
 
kevykevy said:
to radou - basis vectors, example i, j, and k with the carot(^) on top

Ok, that's an example of a basis. We can add that every set consisting of three linearly independent vectors forms a basis for R^3. All you have to do is check if your vectors are linearly independent.
 
  • #10
And if you are going to be doing problems like this it would be a really good idea for you to look at the definition of "basis" in your textbook.
 

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