Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 9: Problem 11
Find the magnitude of the given vector. $$\langle- 2,1,2\rangle$$
Short Answer
Expert verified
The magnitude of the vector is 3.
Step by step solution
01
Understanding the Vector
The vector given is \( \langle -2, 1, 2 \rangle \), which is a vector in three-dimensional space with components \( x = -2 \), \( y = 1 \), and \( z = 2 \). The magnitude of a vector measures its size or length.
02
Formula for Magnitude of a Vector
The magnitude (or length) of a vector \( \langle a, b, c \rangle \) in three-dimensional space is given by the formula \( \sqrt{a^2 + b^2 + c^2} \). We will use this formula to calculate the magnitude of the vector \( \langle -2, 1, 2 \rangle \).
03
Computing the Squares of Components
Calculate each component squared: \( (-2)^2 = 4 \), \( 1^2 = 1 \), and \( 2^2 = 4 \).
04
Summing the Squares
Sum up the squares calculated in the previous step: \( 4 + 1 + 4 = 9 \).
05
Calculating the Square Root
Take the square root of the sum obtained: \( \sqrt{9} = 3 \). This result is the magnitude of the vector.
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.
Three-dimensional vectors
Understanding three-dimensional vectors is crucial in fields like physics and engineering, where they are used to describe spatial relationships and forces. A three-dimensional vector is represented by an ordered triple, such as \( \langle -2, 1, 2 \rangle \).
This notation indicates that the vector has three components, each corresponding to a dimension in space: the x-component, y-component, and z-component.
Key characteristics of three-dimensional vectors:
This notation indicates that the vector has three components, each corresponding to a dimension in space: the x-component, y-component, and z-component.
Key characteristics of three-dimensional vectors:
- They have both a direction and a magnitude (size).
- They are represented graphically as arrows with the tail at the origin and the head at a point in three-dimensional space.
- They can be added or subtracted from each other to yield a resultant vector.
Vector components
Vector components are the building blocks of vectors. In the vector \( \langle -2, 1, 2 \rangle \), each number represents a component of the vector in a particular direction. Here, -2 is the x-component, 1 is the y-component, and 2 is the z-component.
Each component of a vector represents how far the vector points in each respective direction.Understanding vector components involves:
Each component of a vector represents how far the vector points in each respective direction.Understanding vector components involves:
- Identifying each component's contribution to the overall vector.
- Using component notation helps simplify the addition or subtraction of vectors in algebraic operations.
- Knowing the vector components allows one to perform operations such as computing vector magnitude or projecting vectors onto planes.
Vector length
The length or magnitude of a vector is a measure of how long the vector extends in space. It is calculated using the formula for the magnitude of a vector. For a vector \( \langle a, b, c \rangle \), the magnitude is computed using: \( \sqrt{a^2 + b^2 + c^2} \).
This formula comes directly from the Pythagorean theorem, adapted to three dimensions.Steps to find vector length:
This formula comes directly from the Pythagorean theorem, adapted to three dimensions.Steps to find vector length:
- Square each of the components: for the vector \( \langle -2, 1, 2 \rangle \), these are \((-2)^2 = 4\), \(1^2 = 1\), and \(2^2 = 4\).
- Add the squared components together: \(4 + 1 + 4 = 9\).
- Take the square root of the summed values: \(\sqrt{9} = 3\).
Calculus
Calculus is a powerful mathematical field that can be utilized to analyze vectors and their properties. Although the calculation of vector magnitude itself does not directly require calculus, understanding calculus allows for a deeper exploration of vectors in continuous spaces.
In the context of vectors, calculus can help:
- Analyze the behavior of vector fields.
- Determine gradients, which indicate the steepest ascent direction of scalar fields.
- Compute line integrals to find the work done on an object moving along a path in a vector field.
