Introduction
Vector analysis is a branch of mathematics that deals with quantities that have both magnitude and direction. Unlike scalars, which only have magnitude, vectors are essential in representing physical quantities such as force, velocity, and acceleration.
Vector Basics
A vector is often represented as an arrow with its length proportional to the magnitude and direction pointing towards its intended direction. In a coordinate system, a vector in three dimensions can be expressed as:
where , , and are the components of the vector along the x, y, and z axes, respectively, and , , and are the unit vectors along these axes. Unit vectors have length equal to 1.
Vector Operations
Vector addition and subtraction follow the rule of parallelogram law. If and are two vectors, their addition and subtraction are represented as:
Scalar multiplication of a vector scales its magnitude without changing its direction:
where is a scalar.
The dot product (scalar product) and cross product (vector product) are two essential operations in vector analysis.
The dot product of vectors and is:
The cross product of vectors and is:
Applications
Vector analysis is fundamental in physics and engineering fields, notably in mechanics, electromagnetism, and fluid dynamics. It provides a robust mathematical framework to describe and analyze physical phenomena involving directional quantities.
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