Resolving vectors into a unit vector

Similarly, if you already have a vector that is supposed to be a unit vector, you can calculate the corresponding value of N using the same formula. In this case, the unit vector along the antenna polarization is found by taking the sum of the two vectors and dividing it by the square root of the sum of their squares. If there was a factor of 2 added to a_y in the original equation, it would not affect the unit vector result as long as the overall equation still results in a unit vector. In summary, the unit vector along the antenna polarization is found by taking the sum of two vectors and dividing it by the square root of the sum of their squares, and any changes to the original equation will not affect the unit vector result
  • #1
jmc
7
0

Homework Statement


electric field polarization for a given antenna is expressed as:
E_i = (a_x + a_y) E (r, theta, phi)

The unit vector along the antenna polarization is found as u_a = 1/Sqrt(2) (a_x + a_y)2. Question
Where/how is the 1/Sqrt(2) found to resolve those other 2 vectors into a unit vector? Is there a formula or is it some kind of vector addition?
What if there was a factor of 2 added to a_y in the original equation - how would it change that unit vector result?

Thank you
 
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  • #2
A unit vector is defined to have length one. Given any vector ##\vec v## it can be multiplied with ##N = \frac{1}{\sqrt{\vec v^2}}## to give a vector ##\frac{\vec v}{\sqrt{\vec v^2}}## which is of unit length.
 

1. What is a unit vector?

A unit vector is a vector with a magnitude of 1, meaning it has a length of 1 unit. It is used to represent direction without any specific magnitude or scale.

2. How do you find the unit vector of a given vector?

To find the unit vector of a given vector, you divide the vector by its magnitude. This will result in a vector with the same direction as the original, but with a magnitude of 1.

3. Why is it useful to resolve vectors into unit vectors?

Resolving vectors into unit vectors allows us to simplify vector calculations and analysis. It also helps us to focus on the direction of a vector rather than its magnitude.

4. Can a unit vector have a negative magnitude?

No, a unit vector cannot have a negative magnitude. By definition, a unit vector has a magnitude of 1, so it cannot be negative.

5. How do you represent a unit vector mathematically?

A unit vector is represented by a lowercase letter with a hat (^) symbol on top, such as ˆu. This indicates that it is a unit vector. The components of a unit vector can also be represented using the i, j, and k notation, where i represents the x-component, j represents the y-component, and k represents the z-component.

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