Vector addition — the graphical method

Adding vectors is not like adding numbers — magnitudes AND directions must combine. The graphical method is direct, intuitive, and the best way to build first intuition. In §3.4 we’ll see the algebraic (component) method, which is more powerful for precise calculations.

The tip-to-tail rule

To add \vec a + \vec b:

1. Place the tail of \vec b on the tip of \vec a (don’t rotate the vector — just translate it)
2. Draw the sum \vec c = \vec a + \vec b from the tail of \vec a to the tip of \vec b

Physical intuition: think of \vec a as a first displacement and \vec b as a second. Their sum = total displacement. The actual path doesn’t matter — only start and end.

Parallelogram rule — an equivalent method

Alternative:

1. Place the tails of \vec a and \vec b at a common point
2. Draw the parallelogram whose sides are these two vectors
3. The diagonal from the common point = \vec a + \vec b

Both methods (tip-to-tail, parallelogram) give the same result. Tip-to-tail is usually quicker; parallelogram sometimes gives better intuition.

Properties of vector addition

1. Commutative:
$$
\vec a + \vec b = \vec b + \vec a
$$

Order doesn’t matter. Try it — draw \vec a first or \vec b first, the sum is the same.

2. Associative:
$$
(\vec a + \vec b) + \vec c = \vec a + (\vec b + \vec c)
$$

Grouping doesn’t matter. For three or more vectors, group however you like.

3. Identity element:
$$
\vec a + \vec 0 = \vec a
$$

4. Opposite:
$$
\vec a + (-\vec a) = \vec 0
$$

Vector subtraction

We define \vec a - \vec b as addition with the opposite vector:

$$
\vec a – \vec b = \vec a + (-\vec b)
$$

Practically: flip \vec b by 180°, then add to \vec a tip-to-tail.

Interpretation: \vec a - \vec b answers “what must we add to \vec b to get \vec a?” — since \vec b + (\vec a - \vec b) = \vec a.

Example — two forces at an angle

Scenario: two ropes pull a load:
– Rope 1 at 50 N north
– Rope 2 at 50 N east

Total force?

The two vectors are perpendicular. Right triangle. By Pythagoras:

$$
|\vec F_\text{total}| = \sqrt{50^2 + 50^2} = \sqrt{5000} \approx 70.7\ \mathrm{N}
$$

Direction: because both forces are equal and perpendicular, the sum bisects them:

$$
\theta = 45°\ \text{from north toward east (or vice versa)}
$$

Note: simple arithmetic sum (50 + 50 = 100 N) is wrong. Directions matter.

Law of cosines — for arbitrary angles

If two vectors \vec a and \vec b are placed tail-to-tail with angle \phi between them, the magnitude of their sum:

$$
|\vec a + \vec b|^2 = a^2 + b^2 + 2 a b \cos\phi
$$

where a, b are individual magnitudes. Three special cases:

• \phi = 0° (parallel): |\vec a + \vec b| = a + b (max)
• \phi = 90° (perpendicular): |\vec a + \vec b| = \sqrt{a^2 + b^2} (Pythagoras)
• \phi = 180° (antiparallel): |\vec a + \vec b| = |a - b| (min)

A few notes and common mistakes

1. The sum’s magnitude is always between |a-b| and a+b.
Triangle inequality:
$$
|a – b| \leq |\vec a + \vec b| \leq a + b
$$

The closer the angle is to 0°, the larger the sum.

2. Translate vectors parallel to themselves.
When placing \vec b‘s tail on \vec a‘s tip, don’t rotate — only translate.

3. For more than two vectors, chain tip-to-tail.
\vec a + \vec b + \vec c + \vec d = arrow from the very first tail to the very last tip.

4. Only add vectors of the same kind.
You can’t add velocity to force. Same physical dimension only.

What you should be able to do

After this section, you should be able to:

• Draw the tip-to-tail rule to add two vectors
• Use the parallelogram rule as an equivalent method
• Define subtraction as addition of the opposite vector
• Apply the law of cosines for arbitrary angles
• Recognize when the sum is large (small angle) vs small (large angle)
• Understand the triangle inequality for magnitudes

Preview of §3.3

Graphical is great for intuition but inefficient for precise calculation. Try adding 17 vectors at various angles by hand — hopeless. The solution: components. Break each vector into 2 (or 3) scalar numbers along the axes; vector addition becomes scalar addition of corresponding components. §3.3 introduces this decomposition.

📚 See also: Halliday Vol 1, Ch 3, §3.2 — Vector Addition, Geometric Method.
📖 Open reference: OpenStax University Physics Vol 1 — Chapter 2.2: Coordinate Systems and Components.