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Basis vectors

  • Thread starter kevykevy
  • Start date
25
0
1. 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)


2. Homework Equations



3. The Attempt at a Solution
I really have no clue, I think maybe you use either cross product, dot product or triple scalar product...?
 

cristo

Staff Emeritus
Science Advisor
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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.
 

radou

Homework Helper
3,105
6
1. 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?
 

LeonhardEuler

Gold Member
858
1
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 independant, 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?
 

cristo

Staff Emeritus
Science Advisor
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They don't have to be mutually orthogonal to be linearly independant, 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!
 

LeonhardEuler

Gold Member
858
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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:
 
25
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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?
 
25
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to radou - basis vectors, example i, j, and k with the carot(^) on top
 

radou

Homework Helper
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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.
 

HallsofIvy

Science Advisor
Homework Helper
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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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