Find the unit vector \(u_{x}\) in the direction of 1\. \(x=\left[\begin{array}{l}3 \\ 4\end{array}\right]\) 2\. \(x=\left[\begin{array}{c}3 \\ 6 \\ -2\end{array}\right]\) 3\. \(x=\left[\begin{array}{c}1 \\ -1 \\ 1 \\ -1\end{array}\right]\)

The magnitude of a vector is a measure of its length or size. To find the magnitude of a vector, we use the Pythagorean theorem. For a vector with components \(x_1, x_2, x_3, \dots, x_n\), the magnitude \(||x||\) can be calculated using the formula: \[ ||x|| = \sqrt{x_{1}^2 + x_{2}^2 + x_{3}^2 + \cdots + x_{n}^2} \]
This formula sums the squares of each component of the vector and then takes the square root of the result. For example, for the vector \([3,4]\), the magnitude calculation is: \[ ||x|| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
Knowing the magnitude is essential because it helps you understand the vector's scale and is the first step in computing the unit vector.

A unit vector is a vector that has a magnitude of 1 and points in the same direction as the original vector. It is achieved by dividing each component of the original vector by the vector's magnitude. The unit vector \({u_x}\) is given by the formula: \[ u_x = \frac{x}{||x||} \]
This means you take each component of the vector \(x\) and divide it by \(||x||\). For instance, to find the unit vector for \([3, 4]\):
1. Calculate the magnitude: \[ ||x|| = 5 \]
2. Divide each component of the vector by the magnitude: \[ u_x = \frac{1}{5} [3, 4] = [ \frac{3}{5}, \frac{4}{5} ] \]
This process ensures you get a new vector that points in the same direction but has a length of 1. Unit vectors are pivotal in various applications, like directional insights in physics and simplified calculations in linear algebra.

Normalization refers to the process of converting a vector into a unit vector. This involves scaling the vector so that its magnitude is 1 without changing its direction. The formula used is: \[ u_x = \frac{x}{||x||} \]
Here are the steps to normalize a vector:
1. Calculate the magnitude of the vector.
2. Divide each component of the vector by its magnitude.
For a vector like \([3, 6, -2]\), the normalization process involves:
1. Calculating the magnitude: \[ ||x|| = 7 \]
2. Dividing each component by the magnitude: \[ u_x = \frac{1}{7} [3, 6, -2] = [ \frac{3}{7}, \frac{6}{7}, \- \frac{2}{7} ] \]
Normalization is widely used in computer graphics, machine learning, and data processing to ensure uniformity and scale invariance. It helps in making different vectors comparable and simplifies mathematical operations.