Proving the Dot Product and Norm Theorems

Thread starter Design
Start date Sep 30, 2010
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Dot Dot product Norm Product

In summary, the dot product is a mathematical operation that takes two vectors as input and produces a scalar value as output. It is calculated by multiplying the corresponding components of the two vectors and then summing up the results. This can also be calculated using the formula: a · b = |a||b| cos(θ), where a and b are the two vectors, |a| and |b| are their respective magnitudes, and θ is the angle between them. The dot product has various applications in mathematics and physics, including calculating angles between vectors and determining work done by a force. Norm proving, on the other hand, is a mathematical technique used to prove that a given function is a norm, which defines the length or

Sep 30, 2010

Design

Homework Statement

The Attempt at a Solution

I let D be the center x = DX & a = DA

(x-a) * (x+a)=|x|^2-|a|^2

Dunno what to do with the right side of the equation

Last edited: Sep 30, 2010

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Sep 30, 2010

Quinzio

You're just arrived.
The right side is zero, since x = a, and if a dot product of two vectors is zero they are perpendicular.

Sep 30, 2010

Design

x = a since both the radius of the circle correct?

Sep 30, 2010

Quinzio

Yes.

Related to Proving the Dot Product and Norm Theorems

1. What is the dot product?

The dot product, also known as the scalar product, is a mathematical operation that takes two vectors as input and produces a scalar (single) value as output. It is calculated by multiplying the corresponding components of the two vectors and then summing up the results.

2. How is the dot product calculated?

The dot product can be calculated using the formula: a · b = |a||b| cos(θ), where a and b are the two vectors, |a| and |b| are their respective magnitudes, and θ is the angle between them. Alternatively, it can also be calculated by multiplying the x and y components of the vectors and then adding the results.

3. What is the significance of the dot product?

The dot product has various applications in mathematics and physics. It is commonly used to calculate the angle between two vectors, project one vector onto another, and determine the work done by a force in a particular direction. It is also used in computer graphics to calculate lighting and shading effects.

4. What is norm proving?

Norm proving is a mathematical technique used to prove that a given function is a norm, which is a mathematical concept that defines the length or size of a vector. It involves verifying the properties of a norm, such as positivity, homogeneity, and the triangle inequality, for a given function.

5. What is the difference between the dot product and norm proving?

The dot product is a mathematical operation that takes two vectors as input and produces a scalar value as output. On the other hand, norm proving is a mathematical technique used to verify the properties of a given function to determine if it is a norm. They are two different concepts, but the dot product can be used in norm proving to calculate the magnitude of a vector.