Dot product and basis vectors in a Euclidean Space

In summary, The dot product of a vector a with itself can be given by I a I2, which is independent of a basis in a vector space and can be used to define an inner product. This inner product, if not degenerate and positive definite, defines a norm and can be used to do geometry in such spaces. If the basis vectors are not of unit length, then a positive definite, symmetric matrix is needed along with the vectors to set up calculations.
  • #1
Mathematicsresear
66
0

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?
 
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  • #2
Mathematicsresear said:

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.
 
  • #3
fresh_42 said:
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?
 
  • #4
Mathematicsresear said:

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?
 
  • #5
Mathematicsresear said:
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\,!##
 

Related to Dot product and basis vectors in a Euclidean Space

1. What is the dot product in a Euclidean space?

The dot product, also known as the scalar product, is a mathematical operation that takes two vectors in a Euclidean space and returns a scalar value. It is calculated by multiplying the corresponding components of the two vectors and then summing them up. In other words, it measures the similarity or projection of one vector onto another.

2. How is the dot product calculated?

The dot product is calculated by multiplying the corresponding components of the two vectors and then summing them up. For example, if we have two vectors a = (a1, a2, a3) and b = (b1, b2, b3), the dot product would be a · b = a1b1 + a2b2 + a3b3.

3. What is the significance of the dot product in a Euclidean space?

The dot product has several important applications in a Euclidean space. It is used to calculate the angle between two vectors, determine whether two vectors are orthogonal (perpendicular), and find the projection of one vector onto another. It is also used in physics and engineering to calculate work, power, and torque.

4. What are basis vectors in a Euclidean space?

Basis vectors in a Euclidean space are a set of vectors that can be used to represent any vector in that space. They are usually chosen to be orthogonal and have a length of 1, making it easier to calculate the dot product and other operations. The most common basis vectors in a 3-dimensional Euclidean space are i = (1, 0, 0), j = (0, 1, 0), and k = (0, 0, 1), also known as the standard basis.

5. How do basis vectors relate to the dot product?

Since basis vectors are usually chosen to be orthogonal and have a length of 1, the dot product of any two basis vectors would be 0, except for the dot product of a basis vector with itself, which would be 1. This makes it easier to calculate the dot product between any two vectors in a Euclidean space by simply multiplying the corresponding components of the two vectors. Additionally, the basis vectors can be used to find the components of a vector in a specific direction, which is useful in vector operations and calculations.

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