Vectors: Components, Magnitude, and Direction
A plane flies at 200 mph on a heading of 30 degrees north of east while a 40 mph wind pushes from the side. To predict where the plane actually goes, you describe each motion as a vector: a quantity with both a size and a direction. A vector splits cleanly into a horizontal part and a vertical part, and those two numbers carry everything you need.
By the end of this page you can:
- Write a two-dimensional vector in component form and read off its horizontal and vertical parts.
- Compute a vector’s magnitude from its components.
- Recover the components from a given magnitude and direction angle.
The floor. A vector v in the plane is written v = ⟨a, b⟩, where a is the horizontal component and b is the vertical component. Its length is |v| = sqrt(a squared + b squared).
Quick Reference
| Field | Value |
|---|---|
| Textbook | Stewart, Redlin, Watson, Precalculus 7e |
| Chapter | 9 |
| Section | 9.1 Vectors in Two Dimensions |
| Open alternate | OpenStax Precalculus 2e, 8.8 Vectors |
| Difficulty | Core |
| Time | 30 minutes |
Before You Start
The Idea
A vector records a size and a direction at once. In the plane you store that information as two numbers, the component form v = ⟨a, b⟩. The first number a tells you how far the vector reaches horizontally; the second number b tells you how far it reaches vertically. Drawn as an arrow from the origin, the vector ends at the point (a, b).
Two questions go in opposite directions. Given the components, the magnitude is the length of that arrow, which the Pythagorean theorem supplies directly. Given instead a magnitude and a direction angle theta measured counterclockwise from the positive x-axis, the components come from right triangle trigonometry: the horizontal leg is the magnitude times cosine, and the vertical leg is the magnitude times sine.
Key Formulas
For a vector v = ⟨a, b⟩ in the plane, the magnitude is
\[ |v| = \sqrt{a^2 + b^2}. \]
Given a magnitude |v| and a direction angle theta measured from the positive x-axis, the components are
\[ a = |v|\cos\theta, \qquad b = |v|\sin\theta. \]
Source: Stewart 7e, Section 9.1 Vectors in Two Dimensions.
Worked Example
Find the magnitude and direction angle of the vector v = ⟨3, 4⟩.
Step 1. Apply the magnitude formula with a = 3 and b = 4:
\[ |v| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25}. \]
Step 2. Both components are positive, so the vector points into the first quadrant and theta = arctan(b over a) = arctan(4 over 3).
Show the answer
The magnitude is |v| = sqrt(25) = 5.
The direction angle is theta = arctan(4 / 3), which is about 53.13 degrees. Because a = 3 and b = 4 are both positive, the vector lies in the first quadrant, so this value needs no quadrant correction.
A quick check: the components 3 and 4 with magnitude 5 form the familiar 3-4-5 right triangle.
Connections
This builds on right triangle trigonometry, where cosine and sine turn an angle and a hypotenuse into horizontal and vertical legs, and on reference angles, which fix the direction angle when a vector points into the second, third, or fourth quadrant. The component view sets up vector addition and scalar multiplication, where you combine vectors by adding their matching components, and it is the entry point for later work such as the dot product and applications to force and velocity.
Resources
- Stewart, Redlin, Watson, Precalculus 7e, Section 9.1 Vectors in Two Dimensions (primary).
- OpenStax Precalculus 2e, Section 8.8 Vectors (open alternate).
Last updated: 2026-06-24