Section 11.2

Vectors in 3D Space

Extending vector operations to the third dimension using the standard basis .

Introduction

Just as 2D vectors have and components, 3D vectors add a component representing height or depth. Arithmetic operations work exactly the same way: we perform operations component-wise. This lesson focuses on manipulating these 3D arrows algebraically and understanding their geometric behavior.

Interactive: 3D Vector Components

A vector in 3D space.

Key Formulas

Vector Addition

Scalar Multiplication

Parallel Vectors Test

Two vectors are parallel if one is a scalar multiple of the other:

Worked Examples

Example 1: Arithmetic Operations (Level 1)

Given and , find .

Step 1: Scalar Multiply

(Incorporating the negative)

Step 2: Add Components

Example 2: Parallel Vectors (Level 2)

Are and parallel?

Check the ratios of components:

Yes, , so they are parallel (and point in opposite directions).

Example 3: Geometric Application (Level 3)

Find a vector of length 10 in the direction of .

1. Find Magnitude

2. Unit Vector

3. Scale by 10