Section 11.1

Vectors in the Plane

Quantities defined by magnitude and direction, independent of position.

Introduction

In previous courses, we dealt with scalars—single numbers representing quantities like temperature or mass. Now we introduce vectors, which carry two pieces of information: magnitude (length) and direction. This fundamental shift allows us to model forces, velocity, and displacement in multi-dimensional space, serving as the building blocks for all of multivariable calculus.

Visual Concept: Position Independence

Both arrows represent the same vector because they have the same length and direction.

Key Formulas

Vector from Points

From to :

Magnitude (Length)

Distance from tail to head:

Unit Vector

Scales vector to length 1:

Standard Basis

Using and :

Worked Examples

Example 1: Finding a Vector (Level 1)

Find the vector starting at and ending at , and find its magnitude.

Step 1: Subtract Coordinates (End - Start)

Step 2: Calculate Magnitude

Example 2: Finding a Unit Vector (Level 2)

Find a unit vector in the direction of .

Step 1: Magnitude

Step 2: Scale Vector

Divide each component by 5:

The unit vector (green) points the same way but has length 1.

Example 3: Scaled Partial Vector (Level 3)

Find a vector that has magnitude 7 and points in the opposite direction of .

Strategy:

Find unit vector for .
Multiply by -1 to reverse direction.
Multiply by 7 to scale length.

1. Magnitude of v:

2. Unit vector:

3. Result: