Vectors as a Directed Line and Cartesian Components of a Vector

What is a Vector?

A vector is a quantity that has both magnitude and direction. It can be represented as a directed line segment in a plane.

Vectors are often denoted by bold letters, such as $\mathbf{a}$ or $\mathbf{b}$, or with an arrow above the letter, like $\vec{a}$.

Vectors in the Cartesian Plane

In the Cartesian plane, a vector can be described using its Cartesian components. If a vector starts at the origin $(0,0)$ and ends at the point $(x, y)$, its components are $x$ and $y$.

The vector can be written as:

• $\mathbf{a} = \begin{pmatrix} x \\ y \end{pmatrix}$

Magnitude and Direction of a Vector

• Magnitude: The length of the vector, calculated using the formula: $$|\mathbf{a}| = \sqrt{x^2 + y^2}$$
• Direction: The angle $\theta$ the vector makes with the positive x-axis, found using: $$\theta = \tan^{-1}\left(\frac{y}{x}\right)$$

Example

Consider a vector $\mathbf{a}$ with components $(3, 4)$.

Adding and Subtracting Vectors

Vectors can be added or subtracted by adding or subtracting their corresponding components.

• Addition: $\mathbf{a} + \mathbf{b} = \begin{pmatrix} x_1 + x_2 \\ y_1 + y_2 \end{pmatrix}$
• Subtraction: $\mathbf{a} - \mathbf{b} = \begin{pmatrix} x_1 - x_2 \\ y_1 - y_2 \end{pmatrix}$

Example

Given $\mathbf{a} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 1 \\ 4 \end{pmatrix}$, find $\mathbf{a} + \mathbf{b}$ and $\mathbf{a} - \mathbf{b}$.