# 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, πp, that contains the line, r, and it is perpendicular to π.

The plane, πp, 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.