Add and Subtract Vectors

Lesson 4 of 12: VM Cluster

In this lesson:

• Add vectors graphically (head-to-tail and parallelogram rule)
• Add vectors component-wise and in magnitude-direction form
• Subtract vectors as addition of the additive inverse

What You Will Learn Today

1. Add vectors: head-to-tail and parallelogram methods
2. Explain
3. Add vectors component-wise; find the resultant magnitude
4. Add vectors given in magnitude-direction form
5. Define the additive inverse; subtract vectors

Why Isn't Three Plus Four Equal to Seven?

Two forces act on the same bolt:

• Force A: 3 N due east
• Force B: 4 N due north

Total force = 7 N? Let's find out.

Draw what you think the combined force arrow looks like.

Head-to-Tail: Building the Resultant Step by Step

Parallelogram Rule: Both Vectors from One Point

• Place u and w with the same tail
• Complete the parallelogram using dashed lines
• The diagonal from the shared tail = u + w

Both Methods Give the Same Resultant

For u = 3 east and w = 4 north:

• Head-to-tail: slide w to tip of u → resultant from origin to (3,4)
• Parallelogram: diagonal from origin to (3,4)

Worked Example: Sum of Two Force Vectors

N east; N north

Check-In: Estimate the Resultant Graphically

Given: and

1. Draw u and w head-to-tail on a grid
2. Estimate the magnitude of the resultant
3. Is the direction exactly 45°? Why?

Magnitude Inequality: The Triangle Rule

• Same direction: equality holds — no triangle, just a longer segment
• Opposite direction: — partial cancellation
• Any other angle: strictly less than sum, more than difference

Component-Wise Addition: The Algebraic Method

• Horizontal components add independently of vertical components
• Result matches the head-to-tail and parallelogram methods — always

Horizontal motion doesn't interfere with vertical motion — add them separately.

Worked Example: Adding Component Vectors

,

Direction: above positive x-axis.

Adding Vectors in Magnitude-Direction Form

When vectors are given as and , convert to components first:

Three-step process: convert → add components → find resultant magnitude and direction

Special Angles: Adding Vectors in Two Steps

at ; at

General Angles: Convert Then Add Components

at ; at

Check-In: Practice Adding Components Now

Given and :

1. Find
2. Find
3. Find (hint: subtract components)

What Does Subtracting a Vector Mean?

Consider the vector w = .

What is ?

Predict: same magnitude, opposite direction:

Subtracting w is equivalent to adding .

The Additive Inverse and Vector Subtraction

Find and Fix the Magnitude Error

A student wrote:

", : sum has magnitude ."

What is wrong? Compute the correct magnitude and explain.

Full Procedure: Vectors in Magnitude-Direction Form

at ; at

1. Convert each to component form
2. Add component-wise
3. Find the magnitude and direction of the resultant

No intermediate hints provided.

Tip-to-Tip Shortcut for Graphical Subtraction

When v and w share the same tail:

v − w runs from the tip of w to the tip of v

Order matters: tip-of-w to tip-of-v gives v − w; tip-of-v to tip-of-w gives w − v.

Computing Vector Subtraction by Components

,

Magnitude of difference (5) ≠ difference of magnitudes ().

Key Takeaways: Vector Addition and Subtraction

✓ Head-to-tail and parallelogram: two graphical methods, same resultant

✓ Component addition:

✓ Magnitude inequality: — directions interact

✓ Subtraction: ; components subtract coordinate-wise

Next Lesson: Scaling Vectors with Multiplication

You can now add and subtract any two vectors.

Next (VM.B.5): Multiply a vector by a scalar:

Scalar multiplication scales magnitude without changing direction — or reverses it when c < 0.