Intersection Problems

1Find the equation of the plane that passes through the point of intersection between the line Intersecting Line Exercise and the plane Intersecting Plane Exercise and is parallel to the lines:

Intersecting Lines ExerciseIntersecting Lines Exercise

2Find the equation of the line that passes through the point (1, 0, 2) and is parallel to the following lines:

Intersecting Lines Exercise

3Find the value of the parameters a and b for the line Intersecting Lines Exercise if it is coincident with the plane Intersecting Planes Exercise.

4Calculate the values of the parameters a and b so that the following planes pass through the same line:

Intersecting Lines Exercise

5Determine, for different values of a, the relationship (type of intersection) between the following planes:

Intersecting Plane Exercise

6Determine the type of intersection between the plane Intersecting Plane Exercise and the line Intersecting Lines Exercise width= according to different values of the parameter a.

7Determine the value of b so that the line Intersecting Lines Exercise does not cut the plane Plane.

8Calculate the values of m and n so that the linesIntersecting Lines Exerciseand Intersecting Lines Exercise are parallel.

9Calculate the value of k so that the lines Intersecting Lines Exercise and Intersecting Lines Exercise intersect at one point.


1

Find the equation of the plane that passes through the point of intersection between the line Intersecting Lines Problem and the plane Intersecting Plane Problem and is parallel to the lines:

Intersecting Lines ProblemIntersecting Lines Problem

Transform the equation of the line, r, into another equation determined by the intersection of two planes, and these together with the equation of the plane form a system whose solution is the point of intersection.

System

The equation of the plane is determined by the point of intersection and the direction vectors of the parallel lines to the plane.

Linear Determination

Equation of the Plane Solution


2

Find the equation of the line that passes through the point (1, 0, 2) and is parallel to the following lines:

Intersecting Lines Problem

Find a generic point on the line r.

Intersecting Line Calculations

Intersecting Line Calculations

Calculate the equation of the line that passes through Points P and Q.

Intersecting Line Calculations

As the line passes through the point (1, 0, 2), there is:

Intersecting Line Calculations

Intersecting Line Calculations

Solution to the System

Substitute these values into the equation of the line:

Equation of a Line

Operate and simplify.

Equation of the Line Solution


3

Find the value of the parameters a and b for the line Intersecting Lines Problem if it is coincident with the plane Intersecting Plane Problem.

Transform the equation of the line, r, into another equation determined by the intersection of the two planes. These together with the equation of the plane form the system:

System of Equations

Matrices

For the line to be coincident with the plane, the following must be satisfied:

Intersecting Line Calculations

Thus, the determinant of the two matrices of order 3 is equal to zero.

Intersection Solution

Intersection Solution


4

Calculate the values of the parameters a and b so that the following planes pass through the same line:

Intersecting Plane Problem

If the three planes pass through the same line, the following is fulfilled: Range.

System of Equations

Matrices

Intersection Solution

Intersection Solution


5

Determine, for different values of a, the relationship (type of intersection) between the following planes:

Intersecting Planes Problem

Matrices

Determinant

Determinant

Condition

Matrices

The three planes intersect at a point.

Condition

The three equations are identical, thus, the three planes are coincident.

Condition

Matrices

Intersection Solution

Intersection Solution

Since there is no pair of parallel planes, each plane cuts the other two in a line.


6

Determine the type of intersection between the plane Intersecting Plane Problem and the line Intersecting Lines Problem according to different values of the parameter a.

System

Matrices

Determinant

Condition

Range

There is a point intersection.

Condition

Matrices

Determinant

Determinant

There is a line intersection.

Condition

Matrices

Intersection Solution

Intersection Solution

There is no intersection.


7

Determine the value of b so that the line Intersecting Lines Problem does not cut the plane Intersecting Plane Problem.

For the line and the plane to be parallel, the dot product of the direction vector of the line by the normal vector of the plane must be 0.

Intersecting Line Calculations

Scalar Product

Intersection Solution


8

Calculate the values of m and n so that the linesIntersecting Lines Problemand Intersecting Lines Problem are parallel.

If the two lines are parallel, their direction vector must be proportional.

Intersection Solution


9

Calculate the value of k so that the lines Intersecting Lines Problem and Intersecting Lines Problem intersect at one point.

Intersecting Line Calculations

Intersecting Line Calculations

Intersection Solution




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