Lesson 14.2

Vector Arithmetic

Adding vectors, scaling them, and understanding the parallelogram law.

Introduction

Vectors can be added, subtracted, and scaled. These operations follow intuitive geometric rules that match physical reality—like combining forces.

1

Prerequisite Connection

You can write vectors in component form and find their magnitude.

2

Today's Increment

We learn scalar multiplication and vector addition using both algebraic and geometric methods.

3

Why This Matters

Physics uses vector addition constantly: net force, resultant velocity, equilibrium conditions.

Vector Operations

Vector Addition

Add corresponding components

Scalar Multiplication

Multiply each component by c

Vector Subtraction

Add the negative vector

Geometric Interpretation

  • • Head-to-tail: Place tail of at head of . Sum goes from start to end.
  • • Parallelogram law: Both vectors from same point; sum is the diagonal.
  • • Scalar c > 1: Stretches the vector
  • • Scalar 0 < c < 1: Shrinks the vector
  • • Scalar c < 0: Reverses direction

Interactive: Vector Addition (Parallelogram Law)

u = ⟨3, 1⟩+v = ⟨1, 3⟩=u+v = ⟨4, 4⟩

Interactive: Scalar Multiplication

2 × ⟨2, 1⟩ = ⟨4, 2⟩

Worked Examples

Example 1: Adding Vectors

Find where and .

Add corresponding components

Solution

Example 2: Scalar Multiplication

Find and where .

Multiply each component by 3

Multiply each component by -2

Note: The negative reverses direction!

Solution

and

Example 3: Linear Combination

Find where and .

Step 1: Compute 2u

Step 2: Compute 3v

Step 3: Subtract

Solution

Common Pitfalls

Adding magnitudes instead of components

in general. You must add component-by-component first.

Forgetting negative signs in subtraction

For , remember to subtract both components: .

Not distributing scalar to both components

— the scalar multiplies every component.

Real-World Application

Forces in Equilibrium

When multiple forces act on an object at rest, they must sum to zero. If a 100N force pulls east and a 100N force pulls west, the vector sum is .

Engineers use vector addition to ensure bridges, buildings, and machines remain stable under combined loads.

Practice Quiz

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