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Plane Vector Concepts, Linear Operations & Coordinates

Master the fundamentals of plane vectors: concepts, definitions, linear operations (addition, subtraction, scalar multiplication), coordinate representation, and conditions for collinearity and perpendicularity.

Vector Fundamentals
Essential vector concepts and properties
Vector Definition
A mathematical object with both magnitude and direction
a⃗=(x,y)orAB→\vec{a} = (x, y) \quad \text{or} \quad \overrightarrow{AB}
  • Has both magnitude and direction
  • Represented by arrows in geometry
  • Can be expressed in coordinate form
Vector Magnitude
The length or size of a vector
∣a⃗∣=x2+y2|\vec{a}| = \sqrt{x^2 + y^2}
  • Always non-negative
  • Distance formula in coordinate plane
  • For zero vector: |\vec{0}| = 0
Special Vectors
Important vector types with specific properties
Zero: 0⃗Unit: ∣e⃗∣=1\text{Zero: } \vec{0} \quad \text{Unit: } |\vec{e}| = 1
  • Zero vector: magnitude 0, arbitrary direction
  • Unit vector: magnitude 1, specific direction
  • Opposite vectors: \vec{a} = -\vec{b}
Linear Operations
Vector addition, subtraction, and scalar multiplication
Vector Addition
Combining vectors using parallelogram or triangle rule
a⃗+b⃗=(x1+x2,y1+y2)\vec{a} + \vec{b} = (x_1 + x_2, y_1 + y_2)
Key Properties:
  • Commutative: \vec{a} + \vec{b} = \vec{b} + \vec{a}
  • Associative: (\vec{a} + \vec{b}) + \vec{c} = \vec{a} + (\vec{b} + \vec{c})
  • Geometric: head-to-tail placement
Vector Subtraction
Finding difference between vectors
a⃗−b⃗=(x1−x2,y1−y2)\vec{a} - \vec{b} = (x_1 - x_2, y_1 - y_2)
Key Properties:
  • Non-commutative: \vec{a} - \vec{b} \neq \vec{b} - \vec{a}
  • Equivalent to: \vec{a} + (-\vec{b})
  • Geometric: vector from second to first
Scalar Multiplication
Scaling vector magnitude by a real number
λa⃗=(λx,λy)\lambda\vec{a} = (\lambda x, \lambda y)
Key Properties:
  • When λ > 0: same direction
  • When λ < 0: opposite direction
  • When λ = 0: zero vector
Vector Relationships
Conditions for collinearity and perpendicularity
Collinearity (Parallel Vectors)
x1y2−x2y1=0x_1 y_2 - x_2 y_1 = 0
Vectors that lie on the same line or parallel lines
Applications:
  • Check if vectors are parallel
  • Determine if points are collinear
  • Find scalar relationships
Perpendicularity
a⃗⋅b⃗=x1x2+y1y2=0\vec{a} \cdot \vec{b} = x_1 x_2 + y_1 y_2 = 0
Vectors that meet at 90-degree angles
Applications:
  • Check if vectors are perpendicular
  • Find orthogonal directions
  • Verify right angles in geometry
Important Theorems
Fundamental results in vector theory
Plane Vector Basis Theorem
Any vector in a plane can be uniquely expressed as a linear combination of two non-collinear vectors
a⃗=λ1e1⃗+λ2e2⃗\vec{a} = \lambda_1\vec{e_1} + \lambda_2\vec{e_2}
Section Formula
Position vector of a point dividing a line segment in a given ratio
OP→=OA→+λOB→1+λ\overrightarrow{OP} = \frac{\overrightarrow{OA} + \lambda\overrightarrow{OB}}{1 + \lambda}