(a) Prove that the distance from the point (x0, y0)^T to the line a x+b y=0 is (|a x0+b y0|)/(√(

step 1 First problem we want to show that the distance from the origin to the line a x plus b y plus c

(a) Prove that the distance from the point (x0, y0)^T to the...

(a) Prove that the distance from the point $\left(x_0, y_0\right)^T$ to the line $a x+b y=0$ is $\frac{\left|a x_0+b y_0\right|}{\sqrt{a^2+b^2}}$.
(b) What is the minimum distance to the line $a x+b y+c=0$ ?

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Key Concepts

Point-Line Distance Formula

This concept involves the formula that calculates the shortest (perpendicular) distance from a point to a line in the Euclidean plane. The formula d = |ax0 + by0 + c|/?(a² + b²) comes from representing the line as ax + by + c = 0 and measuring how far the given point is from the line along a direction perpendicular to it.

Normal Vector

A normal vector to a line is a vector that is perpendicular to every vector lying along that line. For a line given by ax + by + c = 0, the vector (a, b) serves as a normal vector. This is essential in deriving the distance formula as it provides the direction in which the distance from the point to the line is measured.

Projection and the Dot Product

The shortest distance from a point to a line is found by projecting the vector from any point on the line to the given point onto the normal vector of the line. The dot product is used in this projection process to extract the component of the vector in the direction of the normal vector. Dividing the absolute value of the dot product by the magnitude of the normal vector gives the minimal (perpendicular) distance.

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