Problem 7 Find the magnitude of vector \(\... [FREE SOLUTION] (2024)

Short Answer

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.