In physics, quantities such as displacement, force, and velocity have both magnitude and direction. In mathematics, a quantity with both magnitude and direction is calledvector (vector).
Geometric Representation and Fundamental Concepts of Vectors
We usedirected line segment (directed line segment) to represent vectors. A directed line segment has three elements: start point, direction, and length.
- Vector magnitude: The magnitude of vector $\vec{AB}$ is called the vector's length (or modulus), denoted as $|\vec{AB}|$.
- Special vectors: A vector with zero length is calledzero vector ($\mathbf{0}$); a vector with length 1 is calledunit vector.
- Collinear vectors: Non-zero vectors with the same or opposite directions. It is defined that $\mathbf{0}$ is parallel to any vector.
As long as they have equal magnitude and the same direction, regardless of their starting points, they areequal vectors.
$$\boldsymbol{a} = \boldsymbol{b} \iff |\boldsymbol{a}| = |\boldsymbol{b}| \text{ and same direction}$$
Key Points
Vectors are free: as long as they have equal length and the same direction, they are equal.