Vector Collinearity and Planar Reference Frames

Definition and Properties of Collinear Vectors

Two vectors, denoted as $\mathbf{u}$ and $\mathbf{v}$, are considered to be collinear when there exists a real number $k$ such that the relationship $\mathbf{u} = k\mathbf{v}$ is satisfied. From a geometric perspective, two vectors are defined as collinear whenever they share the same direction.

To demonstrate this concept with a numerical example, consider a vector $\mathbf{u}$ with the coordinates $\begin{pmatrix} 1 \\ 3 \end{pmatrix}$ and a vector $\mathbf{v}$ with the coordinates $\begin{pmatrix} 2 \\ 6 \end{pmatrix}$. By calculating the relationship between these two vectors, it is observed that $2 \times \mathbf{u} = 2 \times \begin{pmatrix} 1 \\ 3 \end{pmatrix} = \begin{pmatrix} 2 \\ 6 \end{pmatrix}$. Because $\mathbf{v} = 2\mathbf{u}$, where $k = 2$ is a real number, it is confirmed that $\mathbf{u}$ and $\mathbf{v}$ are collinear. Conversely, if no such real number exists to relate two vectors, such as the case for vectors $\mathbf{v}$ and $\mathbf{w}$, then those vectors are classified as not being collinear.

Reference Frames in a Plane

The positioning and identification of vectors within a plane are facilitated by a coordinate system known as a frame of the plane. This frame is established using three distinct points in the plane: point $O$, point $I$, and point $J$. The point $O$ serves as the origin, while the vectors formed relative to the other two points, specifically vector $\vec{OI}$ and vector $\vec{OJ}$, define the basis of the plane.

In standard mathematical notation, the vector $\vec{OI}$ is assigned the label $\mathbf{i}$, and the vector $\vec{OJ}$ is assigned the label $\mathbf{j}$. These vectors, $\mathbf{i}$ and $\mathbf{j}$, are formally referred to as unit vectors within the coordinate system. The complete frame of reference is symbolically denoted as $(O; \mathbf{i}, \mathbf{j})$.