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Practice: Intersection of lines
Related Topics
Wize University Linear Algebra Textbook > Equations of Lines and Planes
Intersection of Two Lines
6 Activities
Practice: Intersection of lines
Determine the point(s) of intersection, if any, of the followings lines, or state whether the lines are parallel or skewed.
Part 1
Part 2
Part 3
a)
𝑙
1
:
[
1
,
3
,
4
]
+
𝑡
[
−
1
,
2
,
3
]
𝑙_1:\ \left[1,3,4\right]+𝑡\left[−1,2,3\right]
l
1
:
[
1
,
3
,
4
]
+
t
[
−
1
,
2
,
3
]
𝑙
2
:
[
−
1
,
7
,
10
]
+
𝑠
[
2
,
−
4
,
−
6
]
𝑙_2:\ \left[−1,7,10\right]+𝑠\left[2,−4,−6\right]
l
2
:
[
−
1
,
7
,
10
]
+
s
[
2
,
−
4
,
−
6
]
The lines intersect at exactly one point (1, -2, -3)
The lines intersect at exactly one point (3, -6, -9)
The lines coincide and have infinitely many points of intersection
The lines are parallel and do not intersect
The lines are skewed and do not intersect
I don't know
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More Intersection of Two Lines Questions:
Practice: Intersection of lines
Practice: Intersection of lines
Determine the point(s) of intersection, if any, of the followings lines, or state whether the lines are parallel or skewed.
Practice: Intersection of lines
Practice: Intersection of lines
Determine the point(s) of intersection, if any, of the followings lines, or state whether the lines are parallel or skewed.
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.
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.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 60.6
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
Practice: Intersection of lines
Which of the following statements is/are true about these lines?
L
1
:
[
1
,
1
,
2
]
+
𝑡
[
−
1
,
2
,
3
]
L_1:\ \left[1,1,2\right]+𝑡\left[−1,2,3\right]
L
1
:
[
1
,
1
,
2
]
+
t
[
−
1
,
2
,
3
]
Which of the following statements is true about the lines
L
1
:
x
=
4
+
t
y
=
−
1
+
t
z
=
−
3
−
2
t
L_1:\begin{array}{l}x=4+t\\y=-1+t\\z=-3-2t\end{array}
L
1
:
x
=
4
+
t
y
=
−
1
+
t
z
=
−
3
−
2
t
and
L
2
:
x
=
1
+
r
y
=
2
−
r
z
=
3
−
2
r
L_2:\begin{array}{l}x=1+r\\y=2-r\\z=3-2r\end{array}
L
2
:
x
=
1
+
r
y
=
2
−
r
z
=
3
−
2
r
Find the value of
c
c
c
if the point
(
a
,
b
,
c
)
\left(a,b,c\right)
(
a
,
b
,
c
)
is a point of intersection between the lines
L
1
:
x
=
1
+
t
y
=
t
+
2
z
=
3
−
2
t
L_1:\begin{array}{l}x=1+t\\y=t+2\\z=3-2t\end{array}
L
1
:
x
=
1
+
t
y
=
t
+
2
z
=
3
−
2
t
and
L
2
:
x
=
−
1
+
2
r
y
=
1
+
r
z
=
4
−
r
L_2:\begin{array}{l}x=-1+2r\\y=1+r\\z=4-r\end{array}
L
2
:
x
=
−
1
+
2
r
y
=
1
+
r
z
=
4
−
r
Line Intersections
Consider the lines:
l
1
:
(
x
1
,
x
2
,
x
3
,
x
4
)
=
(
0
,
−
3
,
−
1
,
2
)
+
s
(
1
,
0
,
2
,
−
1
)
l_1:(x_1,x_2,x_3,x_4)=(0,-3,-1,2)+s(1,0,2,-1)
l
1
:
(
x
1
,
x
2
,
x
3
,
x
4
)
=
(
0
,
−
3
,
−
1
,
2
)
+
s
(
1
,
0
,
2
,
−
1
)
l
2
:
(
x
1
,
x
2
,
x
3
,
x
4
)
=
(
6
,
−
4
,
2
,
0
)
+
t
(
−
4
,
1
,
1
,
0
)
l_2:(x_1,x_2,x_3,x_4)=(6,-4,2,0)+t(-4,1,1,0)
l
2
:
(
x
1
,
x
2
,
x
3
,
x
4
)
=
(
6
,
−
4
,
2
,
0
)
+
t
(
−
4
,
1
,
1
,
0
)