# Perpendicular distance from point to a plane

• PirateFan308
In summary, the problem involves finding the perpendicular distance from the point (1,2,3) to the plane x-2y-z=1. One possible solution is to find the equation of a line passing through (1,2,3) that is perpendicular to the plane, and then find the intersection of this line and the plane. Another approach is to use the vector (1,-2,-1), which is known to be perpendicular to the plane, and find the distance between its intersection with the plane and the point (1,2,3). The equation of the line passing through (1,2,3) and perpendicular to the plane can be expressed as x=1+t, y=2-2t,
PirateFan308

## Homework Statement

Find the perpendicular distance from the point (1,2,3) to the plane x-2y-z=1
One method: find the equation of the line throughout (1,2,3) perpendicular to the plane. Find the intersection of this line and the plane

## The Attempt at a Solution

I know the vector (1,-2,-1) is a vector perpendicular to the plane
I'm not sure how to find the distance between where the vector intersects the
plane to the point.

I know what to do if the vector was a line, but I am confused as to how to change the vector into a line.

I think the line for the vector (1,-2,-1) that goes through P (1,2,3) is:
x=1+t y=2-2t z=3-t

Is this correct?

## 1. What is the formula for calculating the perpendicular distance from a point to a plane?

The formula for calculating the perpendicular distance from a point to a plane is d = |ax + by + cz + d| / √(a^2 + b^2 + c^2), where (x, y, z) is the coordinates of the point and ax + by + cz + d = 0 is the equation of the plane.

## 2. How is the perpendicular distance affected by the position of the point relative to the plane?

The perpendicular distance is affected by the position of the point relative to the plane because the distance is measured along a line that is perpendicular to the plane. If the point is located on the plane, the distance is 0. If the point is above the plane, the distance is positive, and if the point is below the plane, the distance is negative.

## 3. Can the perpendicular distance be negative?

Yes, the perpendicular distance can be negative if the point is located below the plane. This means that the point is on the opposite side of the plane compared to the direction of the normal vector.

## 4. What is the significance of calculating the perpendicular distance from a point to a plane?

Calculating the perpendicular distance from a point to a plane is useful in many applications, such as determining the shortest distance between an object and a flat surface. It is also used in geometry and physics to find the distance between two parallel planes or to find the distance from a point to a line in 3D space.

## 5. How is the perpendicular distance related to the concept of orthogonality?

The perpendicular distance from a point to a plane is directly related to the concept of orthogonality because it is the shortest distance between the point and the plane, and it is measured along a line that is perpendicular to the plane. In other words, the perpendicular distance represents the closest possible distance between the point and the plane, making it orthogonal to the plane.

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