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 Equation of the Line Problem and the plane Equation of the Plane Problem, find the equation of the line, s, which is the orthogonal projection of r on π.

4Calculate the distance between the following lines:

Distance Between Lines Problem

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

6Calculate the area of the triangle whose vertices are the points of intersection of the plane Area of a Triangle in Vector Space with the coordinate axes.

7Given the plane Linear Determination 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 Distance distant from the origin and is parallel to the plane Equation of the Plane Exercise.

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:

Area of the Square in Vector Space


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).

Vector Calculations

Vector Calculations

Vectorial Product

Vectorial Operations

Area Solution


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).

Vectorial Operations

Vectorial Operations

Vectorial Operations

Volume Solution


3

Given the line Equation of the Line Problem and the plane Equation of the Plane Exercise, 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.

Perpendicular Planes

Plane

Equation of a Line Solution


4

Calculate the distance between the following lines:

Distance Between Lines Problem

Linear Determination

Linear Determination

Vectorial Operations

Mixed Product

Vectorial Product

Vectorial Operations

Distance Between Lines Solution


5

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

Symmetric Point

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

Perpendicular Line

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

System of Equations

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

Vectorial Operations

Area Solution


6

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

Coordinate Axis

Coordinate Axis

Coordinate Axis

Vectorial Operations

Vectorial Operations

Vectorial Product

Area of a Triangle Solution


7

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

Line and Plane

Linear Determination

Line Equation

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

Perpendicular Line

Coordinate Solution


8

Determine the equation of the plane π that is Root 6 distant from the origin and is parallel to the plane Plane Equation.

Plane Equation

Distance

parámetros

ecuación del plano

distancia


9

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

Equation of a Line

Vectorial Operations

Vectorial Product

Distance Between Lines Solution


10

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

Area of a Square in Vector Space

Line r.

Linear Determination

Line s.

Vectorial Operations

System Solution

Linear Determination


Vectorial Operations

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

Vectorial Product

Distance

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

Area of the Square Solution