Prove that if the line £ is the graph of the linear...

Prove that if the line $£$ is the graph of the linear Cartesian equation $A x+B y+C=0, A^2+B^2 \neq 0$, then all vectors having geometric representations with initial point and terminal point on $\mathfrak{L}$ are parallel. [Hint: Let $\mathbf{S}\left(x_1, y_1\right)$ and $\mathbf{T}\left(x_2, y_2\right)$ be any two points on $\mathfrak{L}$. Show that $A\left(x_2-x_1\right)+B\left(y_2-y_1\right)=0$, and accordingly (Section 1-6) that the vector $\left(x_2-x_1, y_2-y_1\right)$ is perpendicular to the vector $(A, B)$.]

Key Concepts

Dot Product and Orthogonality

The dot product is an algebraic operation that takes two vectors and returns a scalar. A zero dot product indicates that the two vectors are orthogonal (perpendicular) to one another. In the context of the line, showing that the dot product of the displacement vector between two points on the line with the normal vector is zero proves that they are perpendicular, which leads to conclusions about the uniform direction of all such displacement vectors.

Vector Representation of Line Segments

Any segment of the line between two points can be represented by the difference vector obtained by subtracting the coordinates of the initial point from those of the terminal point. This vector captures the direction and magnitude of the displacement along the line and is key to understanding the geometric properties of the line.

Parallelism of Vectors

Parallel vectors share the same or exactly opposite direction. By showing that every vector obtained from two points on the line is perpendicular to the same normal vector, it follows that these vectors must all be parallel to each other. This concept is important in understanding the directional uniformity of lines and translating geometric properties into algebraic proofs.

Cartesian Equation of a Line

A Cartesian equation of a line in two-dimensional space, given by an equation of the form Ax + By + C = 0 (with A^2 + B^2 ? 0), represents all points (x, y) that satisfy the linear relation. This is a fundamental construct in analytic geometry that provides an algebraic description of the geometric object and serves as a basis for further analysis related to direction, intersections, and distances.

Normal Vector

The coefficients (A, B) in the line’s equation define the normal vector to the line. In Euclidean geometry, a normal vector is one that is perpendicular to every vector that lies along the line, and it plays a crucial role in describing the orientation of the line relative to the coordinate axes.

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