Space geometry

1003188904

Level:

Determine the relative position of the plane

ρ

with general equation

7 x - 2 y + z - 2 = 0

and the straight line

p

with parametric equations:

\begin{aligned} x & = 3 + t, \\ y & = - 5 - 2 t, \\ z & = 3 - 11 t; t \in R . \end{aligned}

p ∥ ρ, p ⧸ \subset ρ

p \subset ρ

p

is intersecting the plane

ρ

1003188905

Level:

Determine the relative position of the plane

ρ

with general equation

5 x - 4 y + z - 4 = 0

and the straight line

p

with parametric equations:

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

p

is intersecting

ρ

p ∥ ρ, p ⧸ \subset ρ

p \subset ρ

1003188906

Level:

Let there be planes

α

β

γ

and

δ

defined by their general equations:

\begin{aligned} α : \frac{2}{3} x - 4 y + 6 z - \frac{8}{3} = 0; \\ β : x - 2 y + 3 z - 4 = 0; \\ γ : 2 x - 12 y + 18 z - 4 = 0; \\ δ : x - 6 y + 9 z - 4 = 0. \end{aligned}

Out of the following statements, select the one that is not true.

α ∥ δ, α \neq δ

Planes

β

and

δ

are intersecting.

γ ∥ δ, γ \neq δ

Planes

α

and

β

are intersecting.

α = δ

1003188907

Level:

We are given two intersecting planes

x - 6 y + 9 z - 4 = 0

and

x - 2 y + 3 z - 4 = 0

. Find the parametric equations of their line of intersection

p

\begin{aligned} p : x & = 4, \\ y & = 3 t, \\ z & = 2 t; t \in R \end{aligned}

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

\begin{aligned} p : x & = 4, \\ y & = \frac{3}{2} + 3 t, \\ z & = 1 + 2 t; t \in R \end{aligned}

\begin{aligned} p : x & = 4 + t, \\ y & = \frac{3}{2} + 3 t, \\ z & = 1 + 2 t; t \in R \end{aligned}

1003188908

Level:

Find the real number

k

so that the line

A B

, where

A = [6; k; 5]

and

B = [3; - 1; 2]

, is parallel to the plane

2 x - 3 y + 9 z - 4 = 0

k = 10

k = 12

k = 8

k = 9

1103188705

Level:

Find parametric equations of the line

p

that passes through the point

K = [4; 2; 3]

, is parallel to the

x y

-coordinate plane, and is intersecting the

z

-axis.

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

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

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

\begin{aligned} p : x & = 4 - 2 t, \\ y & = 2 - 4 t, \\ z & = 3 t; t \in R \end{aligned}

1103188706

Level:

We are given points

A = [2; 4; 0]

and

B = [4; 7; 6]

. Find parametric equations of a line

q

, which is the orthogonal projection of the line

A B

into the coordinate plane

x y

\begin{aligned} p : x & = 4 + 2 t, \\ y & = 7 + 3 t, \\ z & = 0; t \in R \end{aligned}

\begin{aligned} p : x & = 2 + 4 t, \\ y & = 4 + 7 t, \\ z & = 6 t; t \in R \end{aligned}

\begin{aligned} p : x & = 4 + 2 t, \\ y & = 7 + 3 t, \\ z & = 6; t \in R \end{aligned}

\begin{aligned} p : x & = 2 - 2 t, \\ y & = 4 - 3 t, \\ z & = - 6 t; t \in R \end{aligned}

1103188804

Level:

Find the standard equation of the plane shown in the picture:

y + 2 z - 4 = 0

x + y + 2 z - 4 = 0

2 y + 4 z = - 8

x + 2 y + 4 z - 8 = 0

1103188805

Level:

Find the standard equation of the plane shown in the picture:

3 x + 5 z - 15 = 0

3 x - 5 z + 15 = 0

3 x + 15 y + 5 z - 15 = 0

3 x - 15 y + 5 z + 15 = 0

1103188806

Level:

Find the standard equation of the plane shown in the picture:

3 x + 2 y - 12 = 0

2 x + 3 y - 26 = 0

2 x + 3 y - z - 12 = 0

3 x + 2 y + z - 12 = 0

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