Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 5: Problem 37
Find the exact magnitude and direction angle to the nearest tenth of a degree of each vector. $$ \langle 3,-1\rangle $$
Short Answer
Expert verified
The magnitude of the vector is approximately 3.2, and the direction angle is approximately 341.6掳.
Step by step solution
01
Finding the Magnitude
To find the magnitude of the vector \ \ \( \langle 3, -1 \rangle \), use the formula for the magnitude of a vector which is \ \ \( \sqrt{x^2 + y^2} \). Here, \( x = 3 \) and \( y = -1 \). \ \ Calculate the magnitude: \ \[ \sqrt{3^2 + (-1)^2} = \sqrt{9 + 1} = \sqrt{10} \approx 3.2 \]
02
Finding the Direction Angle
The direction angle \( \theta \) of the vector can be found using the tangent function \ \ \( \tan \theta = \frac{y}{x} \). \ \ Plugging in the values: \ \[ \tan \theta = \frac{-1}{3} \] \ \ To find the angle, take the inverse tangent (arctan) of \ \( \frac{-1}{3} \): \ \ \( \theta = \tan^{-1} \left( \frac{-1}{3} \right) \) \ \ Use a calculator to find the angle: \ \[ \theta \approx -18.4^{\circ} \]
03
Adjusting the Angle
Since the angle \( \theta \) is negative, it means it is measured clockwise from the positive x-axis. Convert it to a positive angle by adding 360掳 (if necessary, for angles measured from the standard position counterclockwise): \ \[ -18.4^{\circ} + 360^{\circ} = 341.6^{\circ} \]
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Vector Magnitude
Vector magnitude represents the length or size of a vector. For a vector given by coordinates \(\text{\}\langle x, y \rangle\), the magnitude is found using the formula \(\text{\}\sqrt{x^2+y^2}\). This formula is derived from the Pythagorean theorem, which applies to the right triangle formed by the components of the vector on the x- and y-axis. In our example, the vector is \(\text{\}\langle 3, -1 \rangle\), so we calculate the magnitude as:
- \(\text{\}\sqrt{3^2 + (-1)^2} = \text{\}\sqrt{9 + 1} = \text{\}\sqrt{10} \)
Vector Direction
The direction of a vector is the angle it makes with the positive x-axis. This angle helps us understand the exact direction in which the vector points. To find this angle, you need to consider the vector's components (x, y). Using our vector \(\text{\}\langle 3, -1 \rangle\), we need to find the angle by calculating the arctangent of the y-component divided by the x-component. This gives us the direction angle relative to the x-axis.
- The formula is \(\text{\}\theta = \tan^{-1} \left(\frac{y}{x}\right)\)
Tangent Function
The tangent function (\text{tan}) relates the angle of a right triangle to the ratio of its opposite side to its adjacent side. For our vector, the tangent is used to find the angle \(\text{\}\theta \) between the vector and the positive x-axis:
- Using the formula \(\text{\}\tan\theta = \frac{y}{x}\), we substitute our vector components to get: \( \tan\theta = \frac{-1}{3} \)
Inverse Tangent
Inverse tangent, also known as arctangent (\text{tan^{-1}}), is used to find the angle whose tangent is a given number. This is particularly useful when you have a ratio and need to determine the angle. For our vector, we calculate the inverse tangent to find the angle \(\text{\}\theta \):
- \( \theta = \tan^{-1} \left( \frac{-1}{3} \right) \)
- \(-18.4^\text{\}\circ + 360^\text{\}\circ = 341.6^\text{\}\circ \)
Coordinate Vectors
Coordinate vectors, often denoted as < \text{\}\langle x, y \rangle >, define a vector in a two-dimensional space. The first number (x) specifies the vector's horizontal component, while the second number (y) specifies the vertical component. For instance, in our example, the vector \( \text{\}\langle 3, -1 \rangle \), the x-component is 3 and the y-component is -1.
- Coordinate vectors are fundamental in defining positions and directions in a plane.
- They help in performing various vector operations like addition, subtraction, and also finding magnitude and direction.
- When working on a problem, identifying these components correctly is crucial for further calculations.
