# Problems of Distances, Areas and Volumes

### Solutions

1Find the area of the triangle whose vertices are the points A = (1, 1, 1), B = (3, 2, 1) and C = (−1, 3, 2).

2Find the volume of the tetrahedron whose vertices are the points A = (0,0,0), B = (2, 1, 3), C = (−1, 3, 1) and D = (4, 2, 1).

3Given the line and the plane , find the equation of the line, s, which is the orthogonal projection of r on π.

4Calculate the distance between the following lines:

5Find the symmetric point of Point A = (3, 2, 1) to the plane .

6Calculate the area of the triangle whose vertices are the points of intersection of the plane with the coordinate axes.

7Given the plane and the point A = (1, 1, 1), calculate the coordinates of the base (endpoint) of the perpendicular from A to the plane.

8Determine the equation of the plane π that is distant from the origin and is parallel to the plane .

9Find the distance between the point A = (3, 2, 7) and the line of the first octant (+,+,+).

10Calculate the area of the square whose sides are on the lines:

## 1

Find the area of the triangle whose vertices are the points A = (1, 1, 1), B = (3, 2, 1) and C = (−1, 3, 2).

## 2

Find the volume of the tetrahedron whose vertices are the points A = (0,0,0), B = (2, 1, 3), C = (−1, 3, 1) and D = (4, 2, 1).

## 3

Given the line and the plane , find the equation of the line, s, which is the orthogonal projection of r on π.

The line, s, is the intersection of the plane, π, with the plane, π_{p,} that contains the line, r, and it is perpendicular to π.

The plane, π_{p,} is determined by the point A = (2, −1, 0), the vector (2, 1, 1) and the normal vector, (1, 1, 1), of the perpendicular plane π.

## 4

Calculate the distance between the following lines:

## 5

Find the symmetric point of Point A = (3, 2, 1) to the plane .

First, compute r, which is the line that passes through Point A and is perpendicular to π.

Then, find the point of intersection of the line r and the plane π.

Given the coordinates of the midpoint of the line segment, the endpoint A' can be found.

## 6

Calculate the area of the triangle whose vertices are the points of intersection of the plane with the coordinate axes.

## 7

Given the plane and the point A = (1, 1, 1), calculate the coordinates of the base (endpoint) of the perpendicular from A to the plane.

The foot of the perpendicular is the point of intersection between the plane and the line.

## 8

Determine the equation of the plane π that is distant from the origin and is parallel to the plane .

## 9

Find the distance between the point A = (3, 2, 7) and the line of the first octant (+,+,+).

## 10

Calculate the area of the square whose sides are on the lines:

Line r.

Line s.

The distance of r to s is equal to the distance of the point B to the line r.

The side of the square is equal to the distance between the lines r and s.