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Planes in R^3
Related Topics
Wize University Linear Algebra Textbook > Equations of Lines and Planes
Planes in
R
3
\reals^3
R
3
6 Activities
Which one of the following is the standard form equation of the plane in
R
3
\reals^3
R
3
which contains the lines
(
x
,
y
,
z
)
=
(
2
,
0
,
−
1
)
+
t
(
2
,
−
5
,
−
1
)
(x,y,z)=(2,0,-1)+t(2,-5,-1)
(
x
,
y
,
z
)
=
(
2
,
0
,
−
1
)
+
t
(
2
,
−
5
,
−
1
)
and
(
x
,
y
,
z
)
=
(
3
,
1
,
2
)
+
t
(
3
,
−
5
,
1
)
(x,y,z)=(3,1,2)+t(3,-5,1)
(
x
,
y
,
z
)
=
(
3
,
1
,
2
)
+
t
(
3
,
−
5
,
1
)
2
x
+
y
−
z
=
5
2x+y-z=5
2
x
+
y
−
z
=
5
2
x
−
5
y
−
z
=
30
2x-5y-z=30
2
x
−
5
y
−
z
=
30
3
x
−
5
y
+
z
=
30
3x-5y+z=30
3
x
−
5
y
+
z
=
30
x
−
7
y
+
2
z
=
35
x-7y+2z=35
x
−
7
y
+
2
z
=
35
There is no such plane
I don't know
Check Submission
More Planes in
R
3
\reals^3
R
3
Questions:
Write down the equation of a plane defined by this parametric equation: P: (-2,0,5)+s(3,1,2)+t(-2,2,0)
Consider the two lines, L_1:(1,0,0)+s(1,1,1) and L_2:(-1,2,3)+t(4,0,c).
a) For what value of c two lines intersects.
b) Find the equation of the plane containing both lines.
Write down equation of Line L: (1,3,5)+t(-4,1,3) as the intersection of two planes.
Find a point whose distance to the plane
3
x
+
4
y
−
10
z
=
30
3x+4y-10z=30
3
x
+
4
y
−
10
z
=
30
is equal to 10.
Consider two planes with equations
3
x
+
2
y
−
z
=
8
a
n
d
a
x
−
6
y
+
b
z
=
8.
3x+2y-z=8\ and\ ax-6y+bz=8.
3
x
+
2
y
−
z
=
8
an
d
a
x
−
6
y
+
b
z
=
8.
a) For what values of a and b are two planes parallel?
b) For what values of a and b is the angle between two planes 60 degrees?
If two planes
2
x
+
a
y
−
2
z
=
2
y
+
6
2x+ay-2z=2y+6
2
x
+
a
y
−
2
z
=
2
y
+
6
and
x
=
y
+
z
+
3
x=y+z+3
x
=
y
+
z
+
3
are parallel, at which point does the first plane intersect with x-axis.
Consider two lines: L1:(1,0,0)+s(1,1,1) and L2:(-1,2,3)+t(4,0,c).
a) For what value of c two lines intersects.
b) Find the equation of the plane containing both lines.
Find gen form plane thru pt parallel to 2 vecs
Given that the plane
P
P
P
passes through the point
(
1
,
3
,
−
1
)
\left(1,3,-1\right)
(
1
,
3
,
−
1
)
and is parallel to both vectors
𝑢
⃗
=
(
0
,
1
,
1
)
𝑢⃗ = (0,1,1)
u
⃗
=
(
0
,
1
,
1
)
and
𝑣
⃗
=
(
3
,
2
,
1
)
,
\vec𝑣 = (3,2,1),
v
=
(
3
,
2
,
1
)
,
find the standard form equation of the plane.
(type your answer in the form ax+by+cz=d, with no spaces)
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 59.1
Given the points
P
1
=
(
0
,
0
,
1
)
\bcb{P_1 = (0,0,1)}
P
1
=
(
0
,
0
,
1
)
and
P
2
=
(
1
,
−
2
,
3
)
\bcb{P_2 = (1,-2,3)}
P
2
=
(
1
,
−
2
,
3
)
and the vector
v
⃗
=
<
2
,
−
3
,
1
>
\bcb{\vec{v} = \left< 2, -3, 1 \right>}
v
=
⟨
2
,
−
3
,
1
⟩
Find the equation of the plane
Π
\bcb{\Pi}
Π
perpendicular to
v
⃗
\bcb{\vec{v}}
v
, containing the point
P
1
\bcb{P_1}
P
1
.
Planes in $\reals^3$
Practice: Normal Vector
Which of the following vectors is normal to the plane
3
x
+
z
=
7
3x+z=7
3
x
+
z
=
7
?
A plane is parallel to the vector
v
=
⟨
1
,
1
,
0
⟩
v = \langle1, 1, 0\rangle
v
=
⟨
1
,
1
,
0
⟩
. If it passes through points
(
1
,
2
,
−
1
)
(1,2,-1)
(
1
,
2
,
−
1
)
and
(
2
,
1
,
0
)
(2,1,0)
(
2
,
1
,
0
)
then find the equation of the plane.
Which one of the following is a vector that is orthogonal to the line that passes through the points
P
(
1
,
−
1
,
0
)
P\left(1,-1,0\right)
P
(
1
,
−
1
,
0
)
and
Q
(
5
,
0
,
−
3
)
Q\left(5,0,-3\right)
Q
(
5
,
0
,
−
3
)
?
Which of the following is a vector orthogonal/ perpendicular to the plane
2
x
−
z
=
3
2x-z=3
2
x
−
z
=
3
?
133 - FML 3 - 18.1W e.g. 30
Find the equation of a plane passing through the point
(
2
,
0
,
1
)
\bcb{(2, 0, 1)}
(
2
,
0
,
1
)
, perpendicular to the straight line passing through the points
(
1
,
1
,
0
)
\bcb{(1, 1, 0)}
(
1
,
1
,
0
)
and
(
4
,
−
1
,
−
2
)
\bcb{(4, -1, -2)}
(
4
,
−
1
,
−
2
)
.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 60.7
Given the planes
π
1
:
4
x
−
3
y
+
z
=
2
\bcb{\pi_1 \, : \, 4x - 3y + z = 2}
π
1
:
4
x
−
3
y
+
z
=
2
and
π
2
:
x
+
2
y
+
2
z
=
0
\bcb{\pi_2 \, : \, x + 2y + 2z = 0}
π
2
:
x
+
2
y
+
2
z
=
0
, and the lines
L
1
:
{
x
=
3
t
+
2
y
=
−
2
t
+
1
z
=
t
,
L
2
:
{
x
=
−
2
t
+
1
y
=
t
+
1
z
=
t
+
3
,
L
3
:
{
x
=
t
+
3
y
=
−
t
z
=
2
t
+
1
\bcb{L_1 \, : \, \left\{ \begin{matrix} x & = & 3t + 2 \\ y & = & -2t + 1 \\ z & = & t \end{matrix} \right., ~ L_2 \, : \, \left\{ \begin{matrix} x & = & -2t + 1 \\ y & = & t + 1 \\ z & = & t + 3 \end{matrix} \right., ~ L_3 \, : \, \left\{ \begin{matrix} x & = & t + 3 \\ y & = & -t \\ z & = & 2t +1 \end{matrix} \right.}
L
1
:
⎩
⎨
⎧
x
y
z
=
=
=
3
t
+
2
−
2
t
+
1
t
,
L
2
:
⎩
⎨
⎧
x
y
z
=
=
=
−
2
t
+
1
t
+
1
t
+
3
,
L
3
:
⎩
⎨
⎧
x
y
z
=
=
=
t
+
3
−
t
2
t
+
1
,
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 60.4
Given the planes
π
1
:
4
x
−
3
y
+
z
=
2
\bcb{\pi_1 \, : \, 4x - 3y + z = 2}
π
1
:
4
x
−
3
y
+
z
=
2
and
π
2
:
x
+
2
y
+
2
z
=
0
\bcb{\pi_2 \, : \, x + 2y + 2z = 0}
π
2
:
x
+
2
y
+
2
z
=
0
, and the lines
L
1
:
{
x
=
3
t
+
2
y
=
−
2
t
+
1
z
=
t
,
L
2
:
{
x
=
−
2
t
+
1
y
=
t
+
1
z
=
t
+
3
,
L
3
:
{
x
=
t
+
3
y
=
−
t
z
=
2
t
+
1
\bcb{L_1 \, : \, \left\{ \begin{matrix} x & = & 3t + 2 \\ y & = & -2t + 1 \\ z & = & t \end{matrix} \right., ~ L_2 \, : \, \left\{ \begin{matrix} x & = & -2t + 1 \\ y & = & t + 1 \\ z & = & t + 3 \end{matrix} \right., ~ L_3 \, : \, \left\{ \begin{matrix} x & = & t + 3 \\ y & = & -t \\ z & = & 2t +1 \end{matrix} \right.}
L
1
:
⎩
⎨
⎧
x
y
z
=
=
=
3
t
+
2
−
2
t
+
1
t
,
L
2
:
⎩
⎨
⎧
x
y
z
=
=
=
−
2
t
+
1
t
+
1
t
+
3
,
L
3
:
⎩
⎨
⎧
x
y
z
=
=
=
t
+
3
−
t
2
t
+
1
,
The following linear equations represent planes in
R
3
R^3
R
3
. Describe the intersection of these planes (if any).
2
x
+
y
−
z
=
3
x
−
y
−
z
=
3
3
x
+
z
=
6
\begin{array}{c} 2x+y-z=3\\ x-y-z=3\\ 3x+z=6 \end{array}
2
x
+
y
−
z
=
3
x
−
y
−
z
=
3
3
x
+
z
=
6
Planes in $\reals^3$
Practice: Planes in
R
3
\colorOne{\reals^3}
R
3
(Tricky)
Consider the points
A
(
−
1
,
3
,
2
)
A(-1,3,2)
A
(
−
1
,
3
,
2
)
,
B
(
−
3
,
5
,
0
)
B(-3,5,0)
B
(
−
3
,
5
,
0
)
, and
C
(
x
C
,
y
C
,
z
C
)
C(x_C,y_C,z_C)
C
(
x
C
,
y
C
,
z
C
)
, and suppose we know the vector
A
C
→
=
⟨
3
,
−
4
,
−
5
⟩
\overrightarrow{AC}=\lang 3,-4,-5\rang
A
C
=
⟨
3
,
−
4
,
−
5
⟩
.
Find the equation of the plane containing point
C
C
C
, and
perpendicular
to the line passing through points
A
A
A
and
B
B
B
.
133 - FML 3 - 18.1W e.g. 31
Find a point on, and a vector normal to, the plane
2
x
+
y
+
3
z
=
6
\bcb{2x + y + 3z = 6}
2
x
+
y
+
3
z
=
6
.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 57
Find the equation of a plane that contains the three points
P
1
=
(
1
,
−
2
,
3
)
\bcb{P_1 = (1,-2, 3)}
P
1
=
(
1
,
−
2
,
3
)
,
P
2
=
(
0
,
−
1
,
4
)
\bcb{P_2 = (0, -1, 4)}
P
2
=
(
0
,
−
1
,
4
)
and
P
3
=
(
4
,
3
,
−
2
)
\bcb{P_3 = (4,3,-2)}
P
3
=
(
4
,
3
,
−
2
)
.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 50
Find the equation of a plane perpendicular to
w
⃗
=
<
1
,
−
2
,
1
>
\bcb{\vec{w} = \left< 1, -2, 1\right>}
w
=
⟨
1
,
−
2
,
1
⟩
that contains the point
P
=
(
−
1
,
1
,
2
)
\bcb{P = (-1, 1, 2)}
P
=
(
−
1
,
1
,
2
)
.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 59.3
Given the points
P
1
=
(
0
,
0
,
1
)
\bcb{P_1 = (0,0,1)}
P
1
=
(
0
,
0
,
1
)
and
P
2
=
(
1
,
−
2
,
3
)
\bcb{P_2 = (1,-2,3)}
P
2
=
(
1
,
−
2
,
3
)
and the vector
v
⃗
=
<
2
,
−
3
,
1
>
\bcb{\vec{v} = \left< 2, -3, 1 \right>}
v
=
⟨
2
,
−
3
,
1
⟩
, find the point
P
o
\bcb{P_o}
P
o
on the plane
Π
:
2
x
−
3
y
+
z
=
1
\bcb{\Pi: \ 2x - 3y + z = 1}
Π
:
2
x
−
3
y
+
z
=
1
that is closest to the point
P
2
\bcb{P_2}
P
2
.
133 - FML 3 - 18.1W e.g. 31
Find a point on, and a vector normal to, the plane
2
x
+
y
+
3
z
=
6
\bcb{2x + y + 3z = 6}
2
x
+
y
+
3
z
=
6
.