Vector Addition
⭕Definition and Component Form
Component-wise Addition
a + b = (a₁ + b₁, a₂ + b₂, a₃ + b₃, ..., aₙ + bₙ)
2D Vectors
(a₁, a₂) + (b₁, b₂)
= (a₁+b₁, a₂+b₂)
3D Vectors
(a₁, a₂, a₃) + (b₁, b₂, b₃)
= (a₁+b₁, a₂+b₂, a₃+b₃)
Result
Always a vector
Same dimension
🎯Geometric Methods
Triangle Method (Head-to-Tail)
Place the tail of vector b at the head of vector a
- • Draw vector a from origin
- • Draw vector b starting from end of a
- • Resultant: from origin to end of b
- • Most intuitive method
Parallelogram Method
Place both vectors at the same origin
- • Draw both vectors from origin
- • Complete the parallelogram
- • Resultant: diagonal from origin
- • Shows symmetry clearly
⚡Properties of Vector Addition
Commutative
a + b = b + a
Order doesn't matter
Forms same parallelogram
Associative
(a + b) + c = a + (b + c)
Grouping doesn't matter
Can add multiple vectors
Identity Element
a + 0 = a
Zero vector is identity
0 = (0, 0, 0, ...)
📐Vector Subtraction
a - b = a + (-b)
Subtraction is adding the negative
Component Form
a - b = (a₁-b₁, a₂-b₂, a₃-b₃)
Subtract each component
Geometric Meaning
a - b: Vector from b to a
Points from second to first
Reverse of b - a
🔄Scalar Multiplication
Definition
ca = (ca₁, ca₂, ca₃)
Effect: Scales magnitude
Multiply each component by c
Positive Scalar
c > 0
Result: Same direction
c > 1: longer
0 < c < 1: shorter
Negative Scalar
c < 0
Result: Opposite direction
Also scales magnitude
-1: reverses direction
📊Linear Combinations
v = c₁a₁ + c₂a₂ + ... + cₙaₙ
Any vector can be expressed as a combination of basis vectors
Standard Basis (3D)
i = (1, 0, 0)
j = (0, 1, 0)
k = (0, 0, 1)
v = v₁i + v₂j + v₃k
Applications
• Any vector as sum of basis vectors
• Coordinate transformations
• Linear algebra operations
• Vector space theory
📝Unit Vectors and Normalization
û = a / |a|
Unit vector has magnitude 1
Definition
|u| = 1
Length is exactly 1
Direction
Same as original
Only magnitude changes
Usage
Represents direction only
Common in graphics
🔬 Real-World Applications
- • Physics: Force composition and net force
- • Navigation: Course corrections and displacement
- • Computer Graphics: Object transformations and movement
- • Game Development: Character movement and velocity
- • Robotics: Path planning and motion control
- • Engineering: Load distribution and structural analysis
🧮 Problem-Solving Tips
- • Add component by component
- • Use head-to-tail for visualization
- • Break into components for calculation
- • Remember: addition is commutative
- • Use unit vectors for clarity
- • Check dimensions match before adding
- • Result vector starts at origin