Basics of Vectors

Definition

• A vector is an ordered array of numbers that can represent a point in space, direction, or any other mathematical or physical quantity.
• Vectors are typically denoted by bold lowercase letters such as v, u, w.
• The number of elements in a vector determines its dimension. A vector in three-dimensional space might be represented as v = (v1, v2, v3).

Example:

$$\mathbf{v}=\left[\begin{array}{c}{v}_1\\ v_2\\ ⋮\\ v_n\end{array}\right]$$ $$\text{- }{v}_i\text{ denotes the }i\text{-th element of the vector.}$$

---

Vector Norm

• The norm of a vector is a measure of its length or magnitude.
• The norm of vector v is denoted as ||v||, and for a vector in two or three dimensions, it is calculated as the square root of the sum of the squares of its components.

Example:

Consider a 2-dimensional vector $\mathbf{v}=\left[\begin{array}{c}3\\ 4\end{array}\right]$. The norm of $\mathbf{v}$ is:

$$||\mathbf{v}||=\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}=5$$

---

Unit Vector

• A unit vector is a vector of length 1, used to specify a direction.
• It is often obtained by dividing a vector by its norm, resulting in a vector that retains the same direction but with a magnitude of 1.

Example:

Given $\mathbf{v}=\left[\begin{array}{c}3\\ 4\end{array}\right]$, the unit vector $\hat{\mathbf{v}}$ is:

$$\hat{\mathbf{v}}=\frac{\mathbf{v}}{||\mathbf{v}||}=\frac{1}{5}\left[\begin{array}{c}3\\ 4\end{array}\right]=\left[\begin{array}{c}\frac{3}{5}\\ \frac{4}{5}\end{array}\right]$$

---

Zero Vector

• The zero vector is a vector whose elements are all zero.
• It is denoted as 0, and its magnitude (norm) is also zero.
• The zero vector does not have a defined direction.

Example:

$$0=\left[\begin{array}{c}0\\ 0\\ ⋮\\ 0\end{array}\right]$$