/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 7 Find the magnitude of vector \(\... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 7

Find the magnitude of vector \(\mathbf{u}\) if \(\mathbf{u}=\langle- 1,6\rangle\)

Short Answer

Expert verified
The magnitude of the vector is \(\textbf{||u||} = \sqrt{37}\).

Step by step solution

01

Understand the components of the vector

The vector \(\textbf{u} = \langle -1, 6 \rangle\) has components \(-1\) (x-component) and \(6\) (y-component).
02

Recall the formula for magnitude

The magnitude of a vector \(\textbf{u} = \langle a, b \rangle\) is given by the formula: \(\textbf{||u||} = \sqrt{a^2 + b^2}\).
03

Substitute the components into the formula

Substitute \(a = -1\) and \(b = 6\) into the formula: \(\textbf{||u||} = \sqrt{(-1)^2 + 6^2} \).
04

Calculate the squared components

Calculate the squares: \((-1)^2 = 1\) and \(6^2 = 36\).
05

Sum the squared components

Add the squares: \(1 + 36 = 37\).
06

Find the square root of the sum

Take the square root of the sum to find the magnitude: \(\textbf{||u||} = \sqrt{37}\).

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Key Concepts

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

Vector Components
A vector is defined by its direction and magnitude. The components of a vector are essentially its projections along the coordinate axes. For example, in the vector \(\textbf{u} = \langle -1, 6 \rangle\), the number -1 is the x-component, and the number 6 is the y-component.
These components tell us how far the vector goes along each axis. So, \(-1\) means the vector moves one unit to the left on the x-axis, and \6\ means it goes six units up on the y-axis.
Understanding vector components is the first step in finding the magnitude of a vector. Converting the vector into its components helps in applying the magnitude formula correctly.
Magnitude Formula
The magnitude of a vector describes how long the vector is, regardless of its direction. The formula to calculate the magnitude of a vector \(\textbf{u} = \langle a, b \rangle\) is:
\(\textbf{||u||} = \sqrt{a^2 + b^2}\)
This formula comes from the Pythagorean theorem. Its purpose is to measure the length of the vector as if you're finding the hypotenuse of a right triangle formed by the vector's components.
By inserting the components of the vector into the formula, we can calculate its magnitude. For example, with \(\textbf{u} = \langle -1, 6 \rangle\), you would substitute \(-1\) for \a\ and \6\ for \b\.
Square Root Calculation
Finding the magnitude requires calculating the square root of the sum of the squared components. Here's a step-by-step breakdown:
First, square both components of the vector: \(-1)^2 = 1\) and \(6^2 = 36\).
Then, add these squared values together: \(1 + 36 = 37\).
Finally, take the square root of the sum to find the magnitude: \(\textbf{||u||} = \sqrt{37}\).
Using the square root completes the process of finding how long the vector is spatially, ensuring you have accounted for both of its directions.

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