Vector Addition &
Subtraction
Triangle law, parallelogram law, properties, and component-wise addition — master the geometry behind combining vectors.
Unlike scalars, vectors cannot be added by simple arithmetic. Direction must be taken into account. Walking 3 km East and then 4 km North does NOT mean you are 7 km from start — you are only 5 km away (resultant).
Displacement 3 km East (3î) + 4 km North (4ĵ) = 3î + 4ĵ
Resultant magnitude = √(9+16) = 5 km (not 7 km!)
If two vectors a and b are represented by two sides of a triangle taken in the same order, then the third side taken in the reverse order represents a + b.
Note: AB + BC + CA = 0 (sum of triangle vectors = zero)
a = 2î + 3ĵ, b = 4î − ĵ
a + b = (2+4)î + (3−1)ĵ = 6î + 2ĵ
|a + b| = √(36+4) = √40 = 2√10
If two vectors a and b are represented by two adjacent sides of a parallelogram drawn from the same point, then the diagonal from that same point represents a + b.
|a + b| = √(|a|² + |b|² + 2|a||b|cosθ)
Special cases:
θ = 0° → |a + b| = |a| + |b| (max, same direction)
θ = 180° → |a + b| = ||a| − |b|| (min, opposite direction)
θ = 90° → |a + b| = √(|a|² + |b|²)
The resultant magnitude formula is fundamental in JEE Physics and Maths. Also remember: |a − b|² = |a|² + |b|² − 2|a||b|cosθ (angle between a and b for their difference).
2. Associative: (a + b) + c = a + (b + c)
3. Additive Identity: a + 0 = 0 + a = a
4. Additive Inverse: a + (−a) = 0
Board exams frequently ask to prove commutativity and associativity of vector addition using the triangle/parallelogram law. Draw the diagrams clearly with labeled points.
Vector subtraction is defined as a − b = a + (−b). We add the negative of b to a. Geometrically, −b has the same length as b but points in the opposite direction.
a − b = (a₁−b₁)î + (a₂−b₂)ĵ + (a₃−b₃)k̂
a = 5î + 3ĵ − 2k̂, b = 2î − ĵ + 4k̂
a − b = (5−2)î + (3+1)ĵ + (−2−4)k̂ = 3î + 4ĵ − 6k̂
This is the method used in all calculations. Simply add corresponding î, ĵ, k̂ components.
a = î + 2ĵ + 3k̂, b = 4î − ĵ + 2k̂, c = −2î + 3ĵ − k̂
Find a + b + c
= (1+4−2)î + (2−1+3)ĵ + (3+2−1)k̂
= 3î + 4ĵ + 4k̂
|a + b + c| = √(9+16+16) = √41
(Resultant is at most as large as sum of individual magnitudes)
Equality holds when a ∥ b (same direction, θ = 0°)
Also: |a − b| ≥ | |a| − |b| |
Triangle inequality is used in JEE to find bounds on vector magnitudes. If you know |a| and |b|, you can say the resultant lies between | |a|−|b| | and |a|+|b|.