Lesson 7.9

Vector Operations

Add, subtract, and scale vectors using their components. These operations are the foundation for combining forces, velocities, and displacements.

Introduction

Now that you can describe vectors in component form, you can perform arithmetic on them — add components to combine, subtract to find differences, and scale to change magnitude.

Past Knowledge

Vector basics (7.8), component form, magnitude, direction angle.

Today's Goal

Add, subtract, and scalar-multiply vectors in component form.

Future Success

The dot product (7.10) introduces a new operation that measures how aligned two vectors are.

Key Concepts

Operations Summary

Operation	Formula
Addition
Subtraction
Scalar Multiplication

Geometric Interpretation

Addition: place tail of at head of — the resultant goes from first tail to last head. Scalar mult: stretches (or compresses) the arrow. Negative flips direction.

Worked Examples

Basic

Adding Two Vectors

Given: , .

Answer:

Intermediate

Linear Combination

Compute: where and .

Answer:

Advanced

Resultant Force

Two forces: = 50 N at 30°, = 80 N at 135°. Find the resultant.

N at

Answer: ≈ 82.7 N at ≈ 99.3°

Common Pitfalls

Adding Magnitudes Instead of Components

You cannot add magnitudes directly (50 + 80 ≠ 82.7). You must decompose into components first, then add component-wise.

Real-Life Applications

Sports Analytics — Velocity Decomposition

In baseball, a batted ball's velocity is a vector. Analysts decompose it into launch angle (vertical component) and exit velocity direction (horizontal component) to predict whether the ball will be a home run, line drive, or fly out.