Vectors in Two Dimensions

Jamie Z 2025-05-13

Learning Goals

Represent vectors using Cartesian form and polar form.
Convert between Cartesian and Polar forms of vectors.
Calculate the magnitude and direction of a vector.
Perform vector addition and subtraction.
Understand unit vectors and their use.

1. Vector Representation

Vectors have both magnitude (size) and direction. They can be represented in two main ways:

Cartesian Form (Component Form)

A vector is expressed as a column vector: \(\begin{pmatrix} x \ y \end{pmatrix}\), where x and y are the components of the vector along the x-axis and y-axis, respectively.

Represents displacement from the origin to a point (x, y).

Polar Form

A vector is expressed as (r, θ), where:
- r is the magnitude of the vector (distance from the origin).
- θ is the angle the vector makes with the positive x-axis (bearing) measured in degrees.

2. Converting Between Forms

Cartesian to Polar

Magnitude (|v| or r): Use Pythagoras’ theorem: \(r = \sqrt{x^2 + y^2}\)
Direction (θ): Use trigonometry: \(\theta = tan^{-1}(\frac{y}{x})\). Important: Consider the quadrant of the vector to get the correct angle. Your calculator may only give angles between -90° and 90°. Adjust accordingly.

Polar to Cartesian

x-component: x = _r_cos (θ)
y-component: y = _r_sin (θ)

3. Magnitude of a Vector

The magnitude (length) of a vector \(\mathbf{v} = \begin{pmatrix} x \ y \end{pmatrix}\) is calculated as:

\[ |\mathbf{v}| = \sqrt{x^2 + y^2} \]

4. Direction of a Vector

The direction (angle θ) of a vector can be found using the inverse tangent function: Remember to consider the correct quadrant for θ.

\[ \theta = tan^{-1}(\frac{y}{x}) \]

5. Vector Addition and Subtraction

Adding Vectors

To add vectors, add their corresponding components:

If \(\mathbf{a} = \begin{pmatrix} x_1 \ y_1 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} x_2 \ y_2 \end{pmatrix}\), then \(\mathbf{a} + \mathbf{b} = \begin{pmatrix} x_1 + x_2 \ y_1 + y_2 \end{pmatrix}\)

Subtracting Vectors

To subtract vectors, subtract their corresponding components:

If \(\mathbf{a} = \begin{pmatrix} x_1 \ y_1 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} x_2 \ y_2 \end{pmatrix}\), then \(\mathbf{a} - \mathbf{b} = \begin{pmatrix} x_1 - x_2 \ y_1 - y_2 \end{pmatrix}\)

6. Unit Vectors

A unit vector has a magnitude of 1.
To find the unit vector in the direction of a vector v, divide the vector by its magnitude: \(\hat{\mathbf{v}} = \frac{\mathbf{v}}{|\mathbf{v}|} = \begin{pmatrix} \frac{x}{|\mathbf{v}|} \ \frac{y}{|\mathbf{v}|} \end{pmatrix}\)

7. Position Vectors

A position vector describes the location of a point relative to an origin (typically (0,0)).
If A is at coordinates (x, y) and B is at coordinates (x, y), then \(\overrightarrow{AB} = \begin{pmatrix} x_2-x_1 \ y_2 -y_1\end{pmatrix}\)

8. Important Trigonometric Values (for quick reference)

Angle (θ)	sin(θ)	cos(θ)	tan(θ)
0°	0	1	0
30°	½	√3/2	⅓
45°	√2/2	√2/2	1
60°	√3/2	½	√3
90°	1	0	Undefined

Key Reminders:

Always draw a diagram to help visualize the vectors and their directions.
Pay attention to quadrants when finding angles using inverse trigonometric functions.
Be careful with units (ensure consistency).
Use your calculator effectively, especially for inverse trigonometric functions.