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Vector — An Introduction
Definition: A vector is a mathematical object that has both magnitude and direction. Vectors are used across mathematics, physics, engineering, computer graphics, and data science to represent quantities like force, velocity, displacement, and more.
Types and Representations
- Geometric vectors: Shown as arrows in space; length = magnitude, arrowhead = direction.
- Algebraic vectors: Represented as ordered lists of numbers, e.g., in 2D: (x, y); in 3D: (x, y, z).
- Unit vector: A vector with magnitude 1; often used to indicate direction (notation: ˆi, ˆj, ˆk or û).
- Zero vector: All components zero; magnitude 0, no direction.
Basic Operations
- Addition: Component-wise; graphically by the tip-to-tail rule.
- Subtraction: Adding the negative vector.
- Scalar multiplication: Scales magnitude, may reverse direction if scalar is negative.
- Dot product (inner product): a · b = |a||b|cosθ; gives projection and indicates orthogonality (zero means perpendicular).
- Cross product (3D): a × b yields a vector perpendicular to both a and b with magnitude |a||b|sinθ.
Properties & Uses
- Magnitude (length): ||v|| = sqrt(x^2 + y^2 + …).
- Direction cosines/angles: Describe orientation in space.
- Orthogonality and basis: Orthonormal bases simplify decomposition and computation.
- Applications: Physics (forces, motion), computer graphics (transformations, normals), machine learning (feature vectors), robotics (kinematics), signal processing.
Coordinate Systems & Transformations
- Vectors can be expressed in Cartesian, polar (2D), cylindrical or spherical (3D) coordinates.
- Transformations use matrices: scaling, rotation, translation (affine transforms use homogeneous coordinates).
Practical Example (2D)
Given v = (3, 4):
- Magnitude ||v|| = 5
- Unit vector û = (⁄5, ⁄5)
- Dot with w = (1, 0): v · w = 3 (projection onto x-axis)
Tips for Working with Vectors
- Use unit vectors to separate magnitude and direction.
- Normalize before comparing directions.
- For numerical work, watch floating-point precision when vectors are nearly parallel or orthogonal.
- Leverage linear algebra libraries (NumPy, Eigen, GLM) for robust implementations.
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