GeometryInSpace

# Examples

Example 1: Find the plane π and a straight line *s* that pass through the point (5,5,1) and are perpendicular to *r*:

All straight lines in π are perpendicular to *r*, for example:

Example 2: Find the plane π that contains

*r*and*s*:Example 3: Study, depending on *a*, the relative position of these lines:

Example 4: Find the straight line

*t*that passes through the point P(1,0,-1) and intersects_{1}*r*and*s*:Find also the equation of another line, *t _{2}*, perpendicular to both of them.

To obtain *t _{1}* we find the equation of two planes: one with

*r*and

*P*and another with

*s*and

*P*:

To obtain

*t*, we find its direction vector as:_{2}Then, we find the equation of two planes: one with

*r*and*w*and another with*s*and*w*:Example 5: Find the coordinates of the symmetric point of:

- the point

*P*(-1,1,-1) from the line:- the point

*Q*(2,2,-3) from the plane π: 3x – y + 2z – 8 = 0For the first one, we must find the point

*A*. To do that, we find the plane π’ perpendicular to*r*that passes through*P*:For the second one, we must find the point

*B*. To do that, we find the line*s*perpendicular to π that passes through*Q*:**Exercises**:

1.- (PAEG- September 2013)

a) Let the points

b) Calculate the value of λ€

2.- (PAEG- Reserve 1 - 2013) Let the planes: π:

a) Find, in a reasoned way, the values of

b) Find, in a reasoned way, the values of

c) Find, in a reasoned way, the values of

Solutions: 1) a) x + y + 4z = 0; b) λ = 13/5; 2)

*P*(4, 2, 3) and*Q*(2, 0,-5). Find the equation of the plane π that makes*Q*the symmetric point of*P*from the plane π.b) Calculate the value of λ€

**R**so that the plane determined by the points*P, Q*, and*R*(λ,1,0) passes through the origin.2.- (PAEG- Reserve 1 - 2013) Let the planes: π:

*a*x + 2y + z - 4 = 0 (*a*€**R**) and π´: 2x - 4y - 2z -*b*= 0 (*b*€**R**).a) Find, in a reasoned way, the values of

*a*and b that make π and π' coincident planes.b) Find, in a reasoned way, the values of

*a*and*b*that make π and π' parallel and non-coincident planes.c) Find, in a reasoned way, the values of

*a*and*b*that make π and π' perpendicular planes.Solutions: 1) a) x + y + 4z = 0; b) λ = 13/5; 2)

*a*= -1,*b*= -8; b)*a*= -1,*b*≠ -8; c)*a*= 5Obra colocada bajo licencia Creative Commons Attribution Non-commercial Share Alike 3.0 License