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

[kevykevy]

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...?

 

[cristo]

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 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]

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]

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]

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.

 

[kevykevy]

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?

 

[kevykevy]

to radou - basis vectors, example i, j, and k with the carot(^) on top

 

[radou]

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]

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.