Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 7: Problem 8
Find the magnitude and direction angle of each vector. $$ \mathbf{u}=\langle 4,7\rangle $$
Short Answer
Expert verified
Magnitude is \( \sqrt{65} \), direction angle is approximately \( 60.26^\circ \).
Step by step solution
01
Magnitude Formula
To find the magnitude of the vector \( \mathbf{u} = \langle 4, 7 \rangle \), use the formula: \[||\mathbf{u}|| = \sqrt{x^2 + y^2}\]where \( x = 4 \) and \( y = 7 \).
02
Calculate the Magnitude
Substitute \( x = 4 \) and \( y = 7 \) into the magnitude formula: \[||\mathbf{u}|| = \sqrt{4^2 + 7^2} = \sqrt{16 + 49} = \sqrt{65}\]Hence, the magnitude is \( \sqrt{65} \).
03
Direction Angle Formula
To find the direction angle \( \theta \) of the vector, use the arctangent function: \[\theta = \tan^{-1}\left(\frac{y}{x}\right)\]for \( \mathbf{u} = \langle 4, 7 \rangle \), substitute \( x = 4 \) and \( y = 7 \).
04
Calculate the Direction Angle
Substitute \( x = 4 \) and \( y = 7 \) into the direction angle formula:\[\theta = \tan^{-1}\left(\frac{7}{4}\right)\]Calculate to find \( \theta \). This will give you approximately \( 60.26^\circ \).
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 Operations
Vectors are mathematical objects used to represent quantities that have both a magnitude (or size) and a direction. Common examples of vectors in physics include force, velocity, and acceleration. In mathematics, vectors can be depicted in two or more dimensions, such as in the coordinate plane.
- Notation: A vector is often represented as \( \mathbf{u} = \langle x, y \rangle \) for two-dimensional cases, where \( x \) and \( y \) are the vector's components along the axes.
- Vector Addition: To add two vectors, simply add their corresponding components. For example, if \( \mathbf{u} = \langle x_1, y_1 \rangle \) and \( \mathbf{v} = \langle x_2, y_2 \rangle \), then \( \mathbf{u} + \mathbf{v} = \langle x_1 + x_2, y_1 + y_2 \rangle \).
- Scalar Multiplication: Multiply each component of a vector by the scalar value. If \( c \) is a scalar and \( \mathbf{u} = \langle x, y \rangle \), then the product \( c\mathbf{u} = \langle cx, cy \rangle \).
Magnitude Calculation
The magnitude of a vector is a measure of its length, representing the size of the quantity it describes. To calculate the magnitude of a two-dimensional vector, use the Pythagorean theorem.
For a vector \( \mathbf{u} = \langle x, y \rangle \), the magnitude \(||\mathbf{u}||\) is calculated as follows:
For a vector \( \mathbf{u} = \langle x, y \rangle \), the magnitude \(||\mathbf{u}||\) is calculated as follows:
- Formula: \(||\mathbf{u}|| = \sqrt{x^2 + y^2}\)
- \(||\mathbf{u}|| = \sqrt{4^2 + 7^2} = \sqrt{16 + 49} = \sqrt{65}\)
Direction Angle
The direction angle of a vector provides information about its orientation in the plane, specifically indicating which way it is pointing. This angle is typically measured from the positive x-axis, counter-clockwise.
- Direction Formula: To find the angle \( \theta \), use the arctangent function — \( \theta = \tan^{-1}\left(\frac{y}{x}\right)\).
- \( \theta = \tan^{-1}\left(\frac{7}{4}\right) \)
Trigonometric Functions
Trigonometric functions are key tools in calculating vector magnitudes and direction angles. They relate the angles and sides of triangles, playing a pivotal role in vector analysis in both two and three dimensions.
- Sine, Cosine, and Tangent: These functions relate angles to ratios of sides in right-angled triangles.
For a vector \( \mathbf{u} = \langle x, y \rangle \):- Sine represents the ratio of the opposite side to the hypotenuse: \( \sin\theta = \frac{y}{||\mathbf{u}||} \)
- Cosine represents the ratio of the adjacent side to the hypotenuse: \( \cos\theta = \frac{x}{||\mathbf{u}||} \)
- Tangent represents the ratio of the opposite to the adjacent side: \( \tan\theta = \frac{y}{x} \)
- Inverse Functions: These trigonometric functions have inverses we use to calculate angles from ratios: \( \sin^{-1}, \cos^{-1}, \tan^{-1} \).
