Space geometry

9000117408

Level:

In the following list find the plane perpendicular to the plane

ρ

\begin{aligned} ρ : 2 x - 3 y + 7 z - 2 = 0 \end{aligned}

ω : x + 3 y + z + 7 = 0

τ : - 2 x + 3 y - 7 z + 2 = 0

ν : - 2 x - 3 y + 7 z + 2 = 0

σ : 7 x - 3 y + 2 z - 2 = 0

9000117409

Level:

Find the plane parallel to

ρ

passing through the point

M

\begin{aligned} ρ : x - 2 y + 5 z - 3 = 0, M = [3; - 1; 1] \end{aligned}

τ : x - 2 y + 5 z - 10 = 0

σ : 3 x - y + z - 3 = 0

ν : x - 2 y + 5 z + 1 = 0

ω : 3 x - y + z - 11 = 0

9000117410

Level:

Adjust real parameters

p

and

q

to ensure that

ρ

and

σ

are parallel but not identical planes.

\begin{aligned} ρ : 2 x - 3 y + 5 z + 6 = 0, σ : 4 x + p y + q z - 2 = 0 \end{aligned}

p = - 6; q = 10

p = 6; q = 10

p = 6; q = - 10

p = - 6; q = - 10

9000117403

Level:

Determine whether the following planes

ρ

and

σ

are parallel, identical or intersecting.

\begin{aligned} ρ : & x = - u + v, \\ y = u + 2 v, \\ z = - u - v; u, v \in R, \end{aligned} σ : x - 2 y - 3 z + 1 = 0

parallel, not identical

identical

intersecting

9000117401

Level:

Find the intersection of the planes

ρ

and

σ

\begin{aligned} ρ : 2 x - 5 y + 4 z - 10 = 0, σ : x - y - z - 2 = 0 \end{aligned}

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

\begin{aligned} q : x & = 2 s - 10, \\ y & = 5 s - 10, \\ z & = s; s \in R \end{aligned}

\begin{aligned} a : x & = 2 u - 4, \\ y & = 2 u - 4, \\ z & = u; u \in R \end{aligned}

\begin{aligned} b : x & = 3 v + 1, \\ y & = v - 2, \\ z & = v; v \in R \end{aligned}

9000117402

Level:

Determine whether the following planes

ρ

and

σ

are parallel, identical or intersecting.

\begin{aligned} ρ : & x = 2 + u - v, \\ y = 1 + 2 u + 4 v, \\ z = - 1 + 3 u + 3 v; u, v \in R, \end{aligned} \begin{aligned} σ : & x = 2 + r - s, \\ y = 7 + 2 r + 4 s, \\ z = 5 + 3 r + 3 s; s, t \in R . \end{aligned}

identical

parallel, not identical

intersecting

9000117404

Level:

Determine whether the following planes are parallel, identical or intersecting.

\begin{aligned} ρ : \frac{3}{8} x + \frac{1}{2} y - \frac{2}{3} z - 1 = 0, σ : \frac{3}{4} x + y - \frac{4}{3} z - 2 = 0 \end{aligned}

identical

parallel, not identical

intersecting

9000117405

Level:

Determine whether the following planes are parallel, identical or intersecting.

\begin{aligned} ρ : 3 x - y - 4 z + 2 = 0, σ : 6 x - 2 y - 8 z + 5 = 0 \end{aligned}

parallel, not identical

identical

intersecting

9000117406

Level:

Determine whether the following planes are parallel, identical or intersecting.

\begin{aligned} ρ : \frac{3}{2} x - \frac{1}{4} y + \frac{2}{3} z - \frac{2}{5} = 0, σ : \frac{2}{3} x - 4 y + \frac{3}{2} z - \frac{5}{2} = 0 \end{aligned}

intersecting

identical

parallel, not identical

9000117407

Level:

Determine the value of the real parameter

p

which ensures that the following planes are perpendicular.

\begin{aligned} ρ : 2 x - 4 y + 5 z - 4 = 0, σ : - 3 x + p y - 2 z + 4 = 0 \end{aligned}

p = - 4

p = 4

p = 0

p = - 3

9000117408

9000117409

9000117410

9000117403

9000117401

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9000117404

9000117405

9000117406

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