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Intersection of Lines and Planes
Related Topics
Wize University Linear Algebra Textbook > Equations of Lines and Planes
Intersection of a Line and a Plane
5 Activities
Practice: Intersection of Lines & Planes
Find the point of intersection of the line and the plane:
l
:
x
⃗
=
⟨
1
,
2
,
0
⟩
+
t
⟨
2
,
1
,
1
⟩
l: \ \vec x = \lang 1,2,0\rang + t\lang 2,1,1\rang
l
:
x
=
⟨
1
,
2
,
0
⟩
+
t
⟨
2
,
1
,
1
⟩
Π
:
x
+
y
−
3
z
=
0
\Pi: \ x+y−3z=0
Π
:
x
+
y
−
3
z
=
0
A)
(
0
,
3
,
1
)
(0, 3, 1)
(
0
,
3
,
1
)
B)
(
1
,
2
,
1
)
(1, 2, 1)
(
1
,
2
,
1
)
C)
(
2
,
1
,
−
1
)
(2, 1, -1)
(
2
,
1
,
−
1
)
D) Infinitely many points of intersection
E) No intersection
I don't know
Check Submission
More Intersection of a Line and a Plane Questions:
Find the intersection of the line L and the plane P given below
𝐿
:
(
1
,
1
,
0
)
+
𝑡
(
1
,
0
,
−
2
)
𝐿:(1,1,0)+𝑡(1,0,−2)
L
:
(
1
,
1
,
0
)
+
t
(
1
,
0
,
−
2
)
𝑃
:
𝑥
+
𝑦
+
𝑧
=
3
𝑃:𝑥+𝑦+𝑧=3
P
:
x
+
y
+
z
=
3
a) Write the parametric equation of line
L
:
{
x
+
2
y
=
0
x
−
z
=
5
L: \begin{cases} x+2y=0\\ x-z=5 \end{cases}
L
:
{
x
+
2
y
=
0
x
−
z
=
5
b) Find intersection point of the line L with the plane
P
:
3
y
−
3
z
=
2.
P:3y-3z=2.
P
:
3
y
−
3
z
=
2.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 60.3
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. 60.1
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
,
Practice: Intersection of lines and planes
Practice: Intersection of lines and planes
Find the point of intersection of the line
L
:
[
1
,
2
,
0
]
+
𝑡
[
−
1
,
1
,
1
]
L:\ \left[1,2,0\right]+𝑡\left[−1,1,1\right]
L
:
[
1
,
2
,
0
]
+
t
[
−
1
,
1
,
1
]
and the plane
Π
:
𝑥
+
𝑦
−
3
𝑧
=
0.
\Pi:\ 𝑥+𝑦−3𝑧=0.
Π
:
x
+
y
−
3
z
=
0.
Intersection of Lines and Planes
Practice: Intersection of Lines & Planes
Find the point(s) of intersection between the following line and plane:
l
:
{
x
=
1
+
t
y
=
1
−
t
z
=
3
+
2
t
l:\begin{cases} x=1+t\\y=1-t\\z=3+2t \end{cases}
l
:
⎩
⎨
⎧
x
=
1
+
t
y
=
1
−
t
z
=
3
+
2
t