# Determining the Uncertainty of a Unit Vector using Error Propagation

• thefelonwind
In summary, the conversation discusses calculating the unit vector between two points and then using error propagation to determine the uncertainty in the unit vector. The calculated unit vector is (10/sqrt(110)*i + (1/sqrt(110))*j, and the uncertainty is determined by adjusting each coordinate by +/-2 to see the largest possible change in direction.
thefelonwind

## Homework Statement

Consider the points (x, y) = (0,0) and (100,10). Calculate the unit vector u pointing from the first to second. If each coordinate has an uncertainty of +/-2, calculate the uncertainty in u using propagation of error, but making reasonable approximations based on the values given.

## Homework Equations

unit vector = u/||u||

## The Attempt at a Solution

My result for the unit vector is:
unit vector=(10/sqrt(110)*i + (1/sqrt(110))*j
(where i and j represent i-hat and j-hat, the vector components)

I believe this is correct because when I calculate the magnitude of the unit vector using those components, I get 1 as my answer.

Where I am stuck at is calculating the uncertainty in u using error propagation. Can anybody get me on the right track for figuring that out? I've not used error propagation yet (this is for a Introductory Physics Lab) and do not have a single clue where to start.

Welcome to Physics Forums.

thefelonwind said:

## The Attempt at a Solution

My result for the unit vector is:
unit vector=(10/sqrt(110)*i + (1/sqrt(110))*j
(where i and j represent i-hat and j-hat, the vector components)

I believe this is correct because when I calculate the magnitude of the unit vector using those components, I get 1 as my answer.
Yes, that looks good. Also note the i and j parts are in the ratio of 10:1, just as the displacement vector is.

Where I am stuck at is calculating the uncertainty in u using error propagation. Can anybody get me on the right track for figuring that out? I've not used error propagation yet (this is for a Introductory Physics Lab) and do not have a single clue where to start.

I've not seen a problem quite like this one before, but they must mean that there is an error in the angle of the unit vector (the magnitude is exactly 1 by definition).

Can you adjust each given coordinate by ±2, in order to get the direction of the vector to change by the largest possible angle? The resulting unit vector vector you get that way should give an indication of the error.

## What is error propagation?

Error propagation is a method of determining the uncertainty or error in a calculated value that is based on the uncertainties in the individual measurements or variables used to calculate that value.

## What is a unit vector?

A unit vector is a vector with a magnitude of 1 and is used to indicate direction in a coordinate system. It is typically represented by a letter with a hat, such as ɪ̂, ʲ̂, ᴷ̂, to distinguish it from regular vectors.

## Why is determining the uncertainty of a unit vector important?

Determining the uncertainty of a unit vector is important because it allows for a more accurate representation of the direction and magnitude in a measurement or calculation. It also helps to identify and account for any potential errors in the measurement process.

## What factors affect the uncertainty of a unit vector?

The uncertainty of a unit vector is affected by the uncertainties in the individual measurements used to calculate it, as well as the mathematical operations and equations used in the calculation. Additionally, any systematic errors or limitations in the measurement tools or techniques can also impact the uncertainty.

## How is error propagation used to determine the uncertainty of a unit vector?

Error propagation involves using mathematical formulas and techniques to calculate the uncertainty in a calculated value based on the uncertainties in the individual measurements or variables used to calculate it. This can be applied to determine the uncertainty of a unit vector by considering the uncertainties in the vector components and the mathematical operations used to combine them.

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