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Chapter 2: Problem 4

(a) Write a function that returns the two-dimensional unit vector, \(\left(u_{x}, u_{y}\right)\), corresponding to an angle \(\theta\) with the \(x\)-axis. You can use the formula \(\left(u_{x}, u_{y}\right)=\) \((\cos \theta, \sin \theta)\), where \(\theta\) is given in radians. (b) Find the unit vectors for \(\theta=0, \pi / 6, \pi / 3, \pi / 2,3 \pi / 2\). (c) Rewrite the function to instead take the argument \(\theta\) in degrees.

Short Answer

Expert verified
Define functions using \( \cos \) and \( \sin \); find vectors for specified radians; convert degrees to radians for new function.

Step by step solution

01

Write the Function for Radian Input

Define a function that takes an angle \( \theta \) in radians and returns the unit vector \( (u_x, u_y) \). The function uses the formulas \( u_x = \cos(\theta) \) and \( u_y = \sin(\theta) \).
02

Implement the Function

Using the math library in Python (for instance), define the function as follows:```pythonimport mathdef unit_vector_radians(theta): ux = math.cos(theta) uy = math.sin(theta) return (ux, uy)```
03

Find Unit Vectors for Given Angles

Use the defined function to compute the unit vectors for \( \theta = 0 \), \( \theta = \pi / 6 \), \( \theta = \pi / 3 \), \( \theta = \pi / 2 \), and \( \theta = 3\pi / 2 \).
04

Calculate Each Unit Vector

Substitute the given angles into the function:```pythonangles = [0, math.pi/6, math.pi/3, math.pi/2, 3*math.pi/2]unit_vectors = [unit_vector_radians(angle) for angle in angles]```This will output the corresponding unit vectors.
05

Rewrite Function for Degree Input

To rewrite the function for degrees, first convert degrees to radians using the formula \( \theta_{rad} = \theta_{deg} \times \frac{\pi}{180} \). Update the function:```pythondef unit_vector_degrees(theta_deg): theta_rad = math.radians(theta_deg) ux = math.cos(theta_rad) uy = math.sin(theta_rad) return (ux, uy)```

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Unit Vectors in Mechanics
Unit vectors are fundamental in mechanics. They help describe direction without altering magnitude.
To get the unit vector, we use trigonometry. In a 2D plane, we can represent a unit vector with angles using cosine and sine functions.
The formulas are:
\(u_x = \cos(\theta)\)
\(u_y = \sin(\theta)\)
where \(\theta\) is the angle with the x-axis.
This helps break down complex movements into manageable components.
Understanding unit vectors is key in fields such as physics and engineering.
Understanding Trigonometry
Trigonometry is essential to solve the problem. It involves the study of angles and their relationships in triangles.
For our purpose, knowing the cosine and sine functions is crucial.
When dealing with a unit circle:
* Cosine (\(cos(\theta)\)) gives the x-coordinate.
* Sine (\(sin(\theta)\)) gives the y-coordinate.
We leverage these trigonometric functions to break an angle into horizontal and vertical components.
These components help us form the 2D unit vector that only has direction and unit magnitude (1).
Python Programming for Unit Vectors
Python makes it easy to compute unit vectors.
We can define functions and use math libraries for trigonometric calculations.
Here's the function to work with radians:

import math
def unit_vector_radians(theta):
ux = math.cos(theta)
uy = math.sin(theta)
return (ux, uy)
We use the `math` library and basic trigonometric functions to return the unit vector.
Calling this function with different radian values gives various unit vectors quickly.
Radian and Degree Conversion
In trigonometry, angles are crucial. They can be measured in degrees or radians.
Radians are preferred in mathematics and programming.
To convert degrees to radians, we use:
\(\theta_{rad} = \theta_{deg} \times \frac{\pi}{180}\)

We can enhance our function to handle degrees by converting them first.
Here's the modified function:

def unit_vector_degrees(theta_deg):
\ theta_rad = math.radians(theta_deg)
\ ux = math.cos(theta_rad)
\ uy = math.sin(theta_rad)
\ return (ux, uy)
Now, it converts degrees to radians internally using `math.radians` before calculating the unit vector.

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Most popular questions from this chapter

The function \(g(x ; n)\) is given as: $$ g(x ; n)=\frac{\sin (x)}{x^{n}} $$ (a) Make a function gvalue \((\mathrm{x}, \mathrm{n})\) which returns the value of \(g(x ; n)\). (b) Use this function to plot \(\sin (x) / x, \sin (x) / x^{2}\) and \(\sin (x) / x^{3}\) in the same plot for \(-5

The iterative mapping \(x(i+1)=r x(i)(1-x(i))\) is called the logistic map. (a) Make a function logistic \((\mathrm{x}, \mathrm{r})\) which returns the value of \(x(i+1)\) given \(x(i)\) and \(r\) as inputs. (b) Write a script with a loop to calculate the first 100 steps of the logistic map starting from \(x(1)=0.5\). Store all the values in an array \(\mathrm{x}\) with \(n=100\) elements and plot \(x\) as a function of the number of steps \(i\) : (c) Explore the logistic map for \(r=1.0,2.0,3.0\) and \(4.0\).

The file trajectory.dat \({ }^{4}\) contains a list of numbers: to \(x 0 \quad y 0\) t1 \(x 1 y 1\) \(\begin{array}{lll}\cdots & \cdots & \cdots \\ \operatorname{tn} & \mathrm{xn} & \mathrm{yn}\end{array}\) corresponding to the time \(t(i)\) measured in seconds, and the \(x\) and \(y\) positions \(x\) (i) and \(y\) (i) measured in meters for the trajectory of a projectile. (a) Read the data file into the arrays ' \(\mathrm{t}, \mathrm{x}\), and \(y\). (b) Plot the \(x\) and \(y\) positions as function of time in two plots above each other. (c) Plot the \((x, y)\) position of the object in a plot with \(x\) and \(y\) on the two axes.

The normal distribution, often called the Gaussian distribution, is given as: $$ P(x ; \mu, \sigma)=\frac{1}{\sqrt{2 \pi \sigma^{2}}} e^{-(x-\mu)^{2} /\left(2 \sigma^{2}\right)} $$ where \(\mu\) is the average and \(\sigma\) is the standard deviation. (a) Make a function normal \((x, m u\), sigma) that returns the normal distribution value, \(P(x, \mu, \sigma)\) as given by the formula. (b) Use this function to plot the normal distribution for \(-5

You throw a pair of six-sided dice and sum the number from each of the dice: \(Z=X_{1}+X_{2}\), where \(Z\) is the sum of the results from dice 1 , \(X_{1}\), and dice \(2, X_{2} .\) If we perform this experiment many times \((N)\), we can find the average and standard deviation from standard estimators from statistics. The average, \(\langle Z\rangle\), of \(Z\) is estimated from: $$ \langle Z\rangle=\frac{1}{N} \sum_{j=1}^{N} Z_{j} $$ and the standard deviation, \(\Delta Z\), is estimated from: $$ \Delta Z=\frac{1}{N-1}\left(\sum_{j=1}^{N}\left(Z_{j}-\langle Z\rangle\right)^{2}\right. $$ (a) Write a function that returns an array of \(N\) values for \(Z\). (b) Write a function that returns an estimate of the average of an array \(z\) using the formula provided. (c) Write a function that returns an estimate of the standard deviation of an array z using the formula provided. (d) Find the average and standard deviation for \(N=100\) throws of two dice.
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