Addition and Subtraction of Vectors: Explained with Examples

Adding two vectors combines their effects into a single equivalent vector called the resultant. Walk 3 km east and then 4 km north, and your total displacement is one 5 km arrow pointing north-east, not 7 km. The order of the two walks does not matter, but their directions do — that is what makes vector addition different from ordinary addition.

Addition and subtraction matter because nearly every quantity in physics combines this way. Net force is the vector sum of all forces on a body, relative velocity is a vector difference, and navigation, engineering and game physics all depend on getting the resultant right.

Key Concepts

• Triangle law: put b's tail at a's head; a + b runs from a's tail to b's head.
• Parallelogram law: put the tails together and complete the parallelogram; the diagonal from the common tail is a + b — same resultant.
• Components: add matching parts: (a1+b1)î + (a2+b2)ĵ + (a3+b3)k̂.
• Subtraction: a − b = a + (−b); flip b end-for-end, then add.
• Properties: addition is commutative and associative; subtraction is neither, and a − b = −(b − a).

Formulas

• Components: a + b = (a1+b1)î + (a2+b2)ĵ + (a3+b3)k̂; a − b = (a1−b1)î + (a2−b2)ĵ + (a3−b3)k̂. Same coordinates and same units required — displacement cannot join a force.
• Resultant magnitude: |a + b|² = |a|² + |b|² + 2|a||b|cosθ (θ = tail-to-tail angle). θ = 0°: magnitudes add; 180°: subtract; 90°: Pythagoras.
• Difference: |a − b|² = |a|² + |b|² − 2|a||b|cosθ — cosine term's sign flipped.

Worked Example

Let a = 2î + 3ĵ − k̂ and b = î − 2ĵ + 4k̂. Find a + b, a − b and |a + b|.

1. a + b = (2+1)î + (3−2)ĵ + (−1+4)k̂ = 3î + ĵ + 3k̂.
2. |a + b| = √(9 + 1 + 9) = √19 ≈ 4.36.
3. a − b = a + (−b) = (2−1)î + (3+2)ĵ + (−1−4)k̂ = î + 5ĵ − 5k̂.
4. Verify: |a|² = 14, |b|² = 21, so cosθ = (19 − 14 − 21)/(2√294) = −8/√294 ≈ −0.4666, giving θ ≈ 117.8°.
5. Check: 14 + 21 + 2√294×(−8/√294) = 35 − 16 = 19 = |a+b|². Consistent.

Common Misconceptions

• "|a + b| = |a| + |b| always." Only when a and b point the same way. In general the triangle inequality holds: |a + b| ≤ |a| + |b|.
• "a − b means subtracting the lengths." No — it is component-wise addition of the reversed vector, and |a − b| is not |a| − |b| in general.
• "Triangle and parallelogram laws differ." They give the identical resultant; only the construction differs (tails together vs head-to-tail).

Real-World Applications

• Navigation: a ship's true course is the vector sum of its velocity and the ocean current; pilots add wind vectors to airspeed the same way.
• Engineering: tensions in the cables holding a sign are found by requiring their vector sum to cancel the weight exactly.
• Relative motion: rain's velocity relative to a moving car is v_rain − v_car, which is why rain slants toward a moving windshield.

To recap: vectors add head-to-tail by the triangle law (or tail-to-tail by the parallelogram law) and component-wise in coordinates, while subtraction adds the reversed vector. The example gave a + b = 3î + ĵ + 3k̂ and a − b = î + 5ĵ − 5k̂, verified through the cosine formula. Master both the geometry and the components, and resultants stop being guesswork.