Show that 𝐯=a 𝐢+b 𝐣 is perpendicular to the line a x+b y=c...

Show that $\mathbf{v}=a \mathbf{i}+b \mathbf{j}$ is perpendicular to the line $a x+b y=c$ (Hint: For $a$ and $b$ nonzero, establish that the slope of the vector $v$ is the negative reciprocal of the slope of the given line. Also verify the statement when $a=0$ or $\bar{b}=0$

Key Concepts

Line in Standard Form

A line defined by an equation like ax + by = c is expressed in a form where the coefficients a and b determine its orientation and position in the plane. This standard form is useful because it directly reveals properties such as the slope and the normal direction to the line.

Normal Vector

The vector composed of the coefficients (a, b) in the line's equation is known as the normal vector. This vector is perpendicular to every vector that lies along the line, meaning it defines the direction in which the line has no component. Its role is central in understanding the orthogonality between the vector and the line.

Slope and Negative Reciprocal

The slope of a line in the equation ax + by = c is computed as -a/b, reflecting the line’s rate of change in the vertical to horizontal direction. Meanwhile, the vector v = ai + bj has a slope given by b/a (when a is nonzero). Notably, the negative reciprocal of b/a is -a/b, which mathematically shows that the direction of the vector is perpendicular to the line’s direction.

Perpendicularity via Dot Product

Two vectors are perpendicular if their dot product equals zero. In this context, the dot product between any vector parallel to the line and the normal vector (a, b) will be zero, providing algebraic confirmation that the vector v = ai + bj is perpendicular to the line defined by ax + by = c.

Handling Degenerate Cases

Even when one of the coefficients a or b is zero, the notion of a normal vector and the associated perpendicularity remain valid. These special cases, resulting in vertical or horizontal lines, demonstrate that the geometric relationship between the line and its normal vector persists regardless of whether both coefficients are nonzero.

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