Dot product and basis vectors in a Euclidean Space

[Mathematicsresear]

Homework Statement

I am asked to write an expression for the length of a vector V in terms of its dot product in an arbitrary system in Euclidean space.

Homework Equations

The Attempt at a Solution

The dot product of a vector a with itself can be given by I a I². Does that expression only apply for vectors in cartesian coordinates with the standard basis vectors?

 

[fresh_42]

Homework Statement

I am asked to write an expression for the length of a vector V in terms of its dot product in an arbitrary system in Euclidean space.

Homework Equations

The Attempt at a Solution

The dot product of a vector a with itself can be given by I a I². Does that expression only apply for vectors in cartesian coordinates with the standard basis vectors?

No, it is independent of a basis. You can define an inner product (dot product, scalar product) by it's properties alone. Of course, if given a certain vector space, you have to decide somehow whether such a product exists. But you can also just say: Given a vector space with an inner product. Then ... , in which case you won't have to bother whether such a vector space exists or not. The inner product, i.e. if not degenerate and positive definite, defines a norm by $a \cdot a = (a,a)=|a|^2$ which is commonly used as its length (squared). You can also define angles with the help of an inner product. Together this means that we can do geometry in such spaces.

 

[Mathematicsresear]

No, it is independent of a basis. You can define an inner product (dot product, scalar product) by it's properties alone. Of course, if given a certain vector space, you have to decide somehow whether such a product exists. But you can also just say: Given a vector space with an inner product. Then ... , in which case you won't have to bother whether such a vector space exists or not. The inner product, i.e. if not degenerate and positive definite, defines a norm by $a \cdot a = (a,a)=|a|^2$ which is commonly used as its length (squared). You can also define angles with the help of an inner product. Together this means that we can do geometry in such spaces.

What about if I wanted to write an expression for the length of a vector a, given that the basis vectors are not of unit length, how would I proceed?

 

[tnich]

Homework Statement

I am asked to write an expression for the length of a vector V in terms of its dot product in an arbitrary system in Euclidean space.

Homework Equations

The Attempt at a Solution

The dot product of a vector a with itself can be given by I a I². Does that expression only apply for vectors in cartesian coordinates with the standard basis vectors?

I think you can answer that question. What is the length of a vector $\vec V_1 = (r,\theta)$ in polar coordinates? Would applying a dot product to $(r,\theta)$ give you that length?
Do cartesian coordinates need to be stated in terms of standard basis vectors? Suppose you started with a vector $\vec V_2 = x\vec i + y\vec j + z\vec k$ and stated it in terms of an arbitrary basis by applying a linear transformation $A$ to get $\vec V_3 = A \vec V_2$. Under what conditions would $\vec V_2 \cdot \vec V_2 = \vec V_3 \cdot \vec V_3$? What does that tell you about the transformation matrix $A$, and the new basis?

 

[fresh_42]

What about if I wanted to write an expression for the length of a vector a, given that the basis vectors are not of unit length, how would I proceed?

In general you have a positive definite, symmetric matrix $A$ and an inner product $v\cdot w := v^\tau A w$. Now you can do whatever you want to your basis. The matrix $A$ needs one, as well as the vectors if you want to set up calculations. Without them, that's all which is needed. Note that $A$ is not necessary the identity matrix as in the standard inner product in $\mathbb{R}^n\,!$