Intersection Problems

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

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

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

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

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

6Determine the type of intersection between the plane and the line according to different values of the parameter a.

7Determine the value of b so that the line does not cut the plane .

8Calculate the values of m and n so that the linesand are parallel.

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

1

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

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.

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

2

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

Find a generic point on the line r.

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

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

Substitute these values into the equation of the line:

Operate and simplify.

3

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

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:

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

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

4

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

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

5

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

The three planes intersect at a point.

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

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 and the line according to different values of the parameter a.

There is a point intersection.

There is a line intersection.

There is no intersection.

7

Determine the value of b so that the line does not cut the plane .

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.

8

Calculate the values of m and n so that the linesand are parallel.

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

9

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