91Ó°ÊÓ

A vector's magnitude represents its length or size. To find the magnitude of a vector in the form \(\textbf{v} = a \mathbf{i} + b \textbf{j}\), we use the formula: \[ \|\textbf{v}\| = \sqrt{a^2 + b^2} \]. This formula is derived from the Pythagorean Theorem and helps us understand how 'long' the vector is.
For the vector given in the exercise, \(\textbf{v} = -3 \textbf{j}\), the calculation is simple because it only has a y-component. Plugging in the values, we get:
\[ \|\textbf{v}\| = \sqrt{0^2 + (-3)^2} = \sqrt{9} = 3 \].
This gives us the magnitude of \(\textbf{v}\), which is 3.
Remember: The magnitude is always a non-negative number.

Vectors are often broken down into their component form to make operations like addition or finding magnitude easier. A vector in two-dimensional space can be expressed as \(\textbf{v} = a \textbf{i} + b \textbf{j}\), where 'a' represents the x-component and 'b' represents the y-component.

In the exercise, the vector \(\textbf{v} = -3 \textbf{j}\) is already in component form. The x-component is 0, and the y-component is -3. By understanding the components, we can easily apply various calculations, like finding the magnitude or performing vector addition.

Component form simplifies the representation and calculation of vectors, especially in higher dimensions where vectors can have more components.

Vector normalization involves transforming a vector into a unit vector. A unit vector has a magnitude of 1 and retains the same direction as the original vector. The unit vector \(\textbf{u}\) is found by dividing each component of the vector \(\textbf{v}\) by its magnitude \(|\textbf{v}|\).
Using the provided exercise, the given vector is \(\textbf{v} = -3 \textbf{j}\) with a magnitude of 3. We find the unit vector as follows:

\[ \textbf{u} = \frac{\textbf{v}}{|\textbf{v}|} = \frac{-3 \textbf{j}}{3} = -\textbf{j} \].
This results in the unit vector \(\textbf{u}\) being \(-\textbf{j}\), which has a magnitude of 1.
Normalization is important in various applications, from physics to computer graphics, where having a standard length simplifies calculations and analyses.