Space geometry

2010008905

Level:

Determine the relative position of the plane

σ

with general equation

x - 2 y + 3 z - 1 = 0

and the straight line

p

with parametric equations:

\begin{aligned} x & = 4, \\ y & = 5 + 3 t, \\ z & = 2 + 2 t; t \in R . \end{aligned}

p ∥ σ, p ⧸ \subset σ

p \subset σ

p

is intersecting the plane

σ

2010008906

Level:

We are given two intersecting planes

2 x - 3 y + 5 z - 9 = 0

and

3 x - y + 2 z - 1 = 0

. Find the parametric equations of their line of intersection

p

\begin{aligned} p : x & = - 1 - t, \\ y & = - 2 + 11 t, \\ z & = 1 + 7 t; t \in R \end{aligned}

\begin{aligned} p : x & = - 1 - 11 t, \\ y & = - 2 + 11 t, \\ z & = 1 + 7 t; t \in R \end{aligned}

\begin{aligned} p : x & = - 1 + t, \\ y & = - 2 + 11 t, \\ z & = 1 - 11 t; t \in R \end{aligned}

\begin{aligned} p : x & = - 1 - 11 t, \\ y & = - 2 + 11 t, \\ z & = 1 - 11 t; t \in R \end{aligned}

2010008907

Level:

Find the real number

m

so that the line

K L

, where

K = [8; 2; - 2]

and

L = [3; - 2; m]

, is parallel to the plane

4 x - 2 y + 3 z - 5 = 0

m = 2

m = - 2

m = 6

m = - 6

9000101001

Level:

Determine whether two lines

p

and

q

are identical, parallel, intersecting or skew.

\begin{aligned} p : x & = 1 + t, \\ y & = 2 - t, \\ z & = 1 - t; t \in R \end{aligned}

\begin{aligned} q : x & = 2 s, \\ y & = - 1, \\ z & = 2 - 2 s; s \in R \end{aligned}

intersecting lines

skew lines

identical lines

parallel lines, not identical

9000101002

Level:

Find the intersection of the line

A B

and the line

p

, where

A = [0; 1; 2]

B = [4; 1; - 2]

and

\begin{aligned} p : x & = 1 + t, \\ y & = 2 - t, \\ z & = 1 - t; t \in R . \end{aligned}

[2; 1; 0]

[1; 2; 1]

[3; 0; - 1]

There is no intersection.

9000101003

Level:

Find the value of the real parameter

m \in R

which ensures that the lines

p

and

q

are parallel and not identical.

\begin{aligned} p : x & = 1 + t, \\ y & = 2 - t, \\ z & = 1 - t; t \in R \end{aligned} \begin{aligned} q : x & = s, \\ y & = - s, \\ z & = 3 + m s; s \in R . \end{aligned}

m = - 1

m = - 2

m = 0

m = 1

9000101004

Level:

Find all the values of the real parameter

m

so that the lines

p

and

q

are skew lines.

\begin{aligned} p : x & = 1 + t, \\ y & = 2 - t, \\ z & = 1 - t; t \in R \end{aligned} \begin{aligned} q : x & = s, \\ y & = 1 + s, \\ z & = 3 + m s; s \in R \end{aligned}

m \in R ∖ {- 2}

No solution exists.

The lines are skew for every real

m

m = - 2

9000101005

Level:

Find the value of the real parameter

m

which ensures that the lines

p

and

q

are intersecting lines (with a unique common point).

\begin{aligned} p : x & = 1 + t, \\ y & = 2 - t, \\ z & = 1 - t; t \in R \end{aligned} \begin{aligned} q : x & = s, \\ y & = 1 + s, \\ z & = 3 + m s; s \in R \end{aligned}

m = - 2

No solution exists.

The lines are intersecting for every real

m

m = 2

9000101006

Level:

Find the value of the real parameter

m

which ensures that the following lines are parallel and not identical lines.

\begin{aligned} p : x & = 1 + t, \\ y & = 2 - t, \\ z & = 1 - t; t \in R \end{aligned} \begin{aligned} q : x & = s, \\ y & = 1 + s, \\ z & = 3 + m s; s \in R \end{aligned}

No solution exists.

The lines are parallel and not identical for every real

m

m = - 2

m = 2

9000101007

Level:

Find the value of the real parameter

m

which ensures that the following two lines are identical.

\begin{aligned} p : x & = 1 + t, \\ y & = 2 - t, \\ z & = 1 - t; t \in R \end{aligned} \begin{aligned} q : x & = s, \\ y & = 1 + s, \\ z & = 3 + m s; s \in R \end{aligned}

No solution exists.

The lines are identical for every real

m

m = - 2

m = 2

2010008905

2010008906

2010008907

9000101001

9000101002

9000101003

9000101004

9000101005

9000101006

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