Learn To Calculate The Shortest Distance Between A Point And A Line In Math: Vector Projection And Distance Formula Explained

distance from a point to a line

The length of the perpendicular segment from the point to the line.

The distance from a point to a line is the shortest distance between the point and any point on the line. To find the distance, we can use the formula:

distance = |ax + by + c| / √(a² + b²)

where (a, b) is the direction vector of the line, c is a constant term, and (x, y) are the coordinates of the point.

Alternatively, we can use vector projection to find the distance:

distance = |(p – q) · n| / |n|

where p is the point, q is any point on the line, n is the direction vector of the line, and · denotes dot product.

To illustrate the process with an example, let’s say we want to find the distance from the point P(2, -1) to the line L given by the equation 2x + y – 1 = 0.

First, we need to find a point Q on the line. We can do this by choosing one of the variables and solving for the other. Let’s solve for y:

2x + y – 1 = 0
y = 1 – 2x

So if we set x = 0, we get y = 1, which gives us the point Q(0, 1).

Next, we need to find the direction vector of the line. We can do this by looking at the coefficients of x and y in the equation. The direction vector is (2, 1).

Now we can use the formula to find the distance:

distance = |2(2) + 1(-1) – 1| / √(2² + 1²)
distance = 3 / √5

So the distance from P to L is 3/√5 units.

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