Vector addition and subtraction

Adding vectors

Vector addition combines vectors to find their total effect: the resultant.

Method 1: Tip-to-tail

1. Draw the first vector.
2. Place the tail of the second vector at the head of the first.
3. The resultant goes from the tail of the first to the head of the last.

This tip-to-tail method works for any number of vectors.

Method 2: Parallelogram

1. Draw both vectors from the same starting point.
2. Complete the parallelogram.
3. The diagonal from the common starting point is the sum.

The parallelogram method shows that u + v = v + u.

Method 3: Components (most precise)

Add the components separately.

[a, b] + [c, d] = [a + c, b + d]

Example: [3, 4] + [5, −1] = [8, 3]

Adding vectors at angles

Example: A hiker walks 5 km east, then 3 km north. Resultant?

• [5, 0] + [0, 3] = [5, 3]
• Magnitude: √34 ≈ 5.83 km
• Direction: tan⁻¹(3/5) ≈ 31.0° north of east

Example: 40 N at 0° plus 30 N at 60°

• First: [40, 0]
• Second: [30 cos 60°, 30 sin 60°] = [15, 25.98]
• Sum: [55, 25.98]
• Magnitude ≈ 60.8 N at ≈ 25.3°

Alternatively, use the cosine law in the vector triangle.

Vector subtraction

u − v = u + (−v): reverse v and add.

[a, b] − [c, d] = [a − c, b − d]

Example: [7, 2] − [3, −4] = [4, 6]

Geometrically, u − v goes from the head of v to the head of u (when both start at the same point).

Properties

Applications

• Forces: net force on an object
• Navigation: a plane’s velocity plus wind velocity
• Displacement: several legs of a journey

Example: A plane flies at 250 km/h north; wind blows 50 km/h east. Ground velocity: [50, 250] → √65,000 ≈ 255 km/h, about 11.3° east of north.

Worked examples

Example 1: [−2, 5] + [6, −1] = [4, 4].

Example 2: [1, 2, 3] + [4, −2, 0] = [5, 0, 3].

Example 3: If u + v = 0, how are u and v related? Opposite vectors.

Common mistakes

• Adding magnitudes instead of vectors.
• Placing vectors tail-to-tail in the tip-to-tail method.
• Subtracting in the wrong order.

Try it yourself

1. [4, −3] + [−1, 7]
2. [5, 2] − [8, −1]
3. Walk 6 km north, then 8 km east. Resultant magnitude?
4. Forces of 10 N east and 10 N north. Resultant?

Answers: 1) [3, 4] 2) [−3, 3] 3) 10 km 4) ≈ 14.1 N at 45° (northeast)

Quick recap

• Vector addition gives a resultant: use the tip-to-tail method, the parallelogram method or components.
• Vector subtraction adds the opposite.
• Add components separately for precise answers.