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Equation of Lines
Related Topics
Wize University Linear Algebra Textbook > Equations of Lines and Planes
Lines in
R
n
\reals^n
R
n
6 Activities
Which of of the following is an equation of the line that passes through
A
(
0
,
−
3
)
A(0,-3)
A
(
0
,
−
3
)
and
B
(
−
3
,
6
)
B(-3,6)
B
(
−
3
,
6
)
.
(
x
,
y
)
=
(
0
,
−
3
)
+
(
3
,
1
)
(x,y)=(0,-3)+(3,1)
(
x
,
y
)
=
(
0
,
−
3
)
+
(
3
,
1
)
(
x
,
y
)
=
(
1
−
t
)
(
1
,
−
3
)
+
t
(
3
,
1
)
(x,y)=(1-t)(1,-3)+t(3,1)
(
x
,
y
)
=
(
1
−
t
)
(
1
,
−
3
)
+
t
(
3
,
1
)
3
x
+
y
=
−
3
3x+y=-3
3
x
+
y
=
−
3
x
−
3
y
=
9
x-3y=9
x
−
3
y
=
9
(
−
3
,
1
)
⋅
(
(
x
,
y
)
−
(
0
,
−
3
)
)
=
0
(-3,1)\cdot((x,y)-(0,-3))=0
(
−
3
,
1
)
⋅
((
x
,
y
)
−
(
0
,
−
3
))
=
0
I don't know
Check Submission
More Lines in
R
n
\reals^n
R
n
Questions:
Consider u=(1,2,3) and v=(2,-1,4)
a. Find a parametric equation of the line passing through the point (1,3,1) and
perpendicular to both
u
and
v.
1) Find a vector of length 1 that is parallel to the line given by the equations:
𝑦
−
𝑧
=
4
2
𝑥
+
𝑦
+
3
𝑧
=
2
\begin{array}{}𝑦−𝑧=4\\2𝑥+𝑦+3𝑧=2 \end{array}
y
−
z
=
4
2
x
+
y
+
3
z
=
2
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.
Find vec form line thru pt ortho to 2 vec form lines
Find the equation of the line that passes through the point
(
1
,
2
,
−
3
)
\left(1,2,-3\right)
(
1
,
2
,
−
3
)
and is orthogonal to both lines
l
1
:
(
−
5
,
0
,
1
)
+
t
(
1
,
0
,
−
1
)
l_1:\ \left(-5,\ 0,\ 1\right)+t\left(1,0,-1\right)
l
1
:
(
−
5
,
0
,
1
)
+
t
(
1
,
0
,
−
1
)
and
l
2
:
(
3
,
−
1
,
6
)
+
t
(
0
,
2
,
1
)
l_2:\ \left(3,-1,6\right)+t\left(0,2,1\right)
l
2
:
(
3
,
−
1
,
6
)
+
t
(
0
,
2
,
1
)
.
Practice Question: Two Point Form Equation
Write a vector that is parallel to the line
(
x
,
y
,
z
)
=
(
1
−
t
)
(
6
,
−
1
,
2
)
+
t
(
9
,
5
,
−
7
)
\left(x,\ y,\ z\right)=\left(1-t\right)\left(6,-1,2\right)+t\left(9,5,-7\right)
(
x
,
y
,
z
)
=
(
1
−
t
)
(
6
,
−
1
,
2
)
+
t
(
9
,
5
,
−
7
)
.
Find vec form line thru pt ortho to 2 vec form lines
Find the equation of the line that passes through the point
(
1
,
2
,
−
3
)
\left(1,2,-3\right)
(
1
,
2
,
−
3
)
and is orthogonal to the lines with equations:
(
x
,
y
,
z
)
=
(
−
5
,
0
,
1
)
+
t
(
1
,
0
,
−
1
)
(x,y,z)= (-5,0,1)+t(1,0,-1)
(
x
,
y
,
z
)
=
(
−
5
,
0
,
1
)
+
t
(
1
,
0
,
−
1
)
and
(
x
,
y
,
z
)
=
(
3
,
−
1
,
6
)
+
t
(
0
,
2
,
1
)
(x,y,z)=(3,-1,6)+t(0,2,1)
(
x
,
y
,
z
)
=
(
3
,
−
1
,
6
)
+
t
(
0
,
2
,
1
)
Which of the following vectors is a normal vector of a line that is parallel to
2
x
+
y
=
−
3
2x+y=-3
2
x
+
y
=
−
3
?
133 - FML 3 - 18.1W e.g. 33; 222/262_mid_#2_19.1W_eg5
Find the equations of the line passing through the point
(
2
,
3
,
−
4
)
\bcb{(2, 3, -4)}
(
2
,
3
,
−
4
)
parallel to the vector
i
−
4
k
\bcb{\mathbf{i} - 4\mathbf{k}}
i
−
4k
. Express them in vector-parametric, scalar-parametric and standard form.
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
)
.
Lines in $\reals^n$
Practice: Lines in
R
n
\colorOne{\reals^n}
R
n
Various pieces of information can be given to find the equation of a line.
133 - FML 3 - 18.1W e.g. 58.3
Given the vector
v
⃗
=
<
1
,
1
,
2
>
\bcb{\vec{v} = \left< 1, 1, 2 \right>}
v
=
⟨
1
,
1
,
2
⟩
and the points
P
1
=
(
0
,
−
1
,
0
)
\bcb{P_1 = (0,-1,0)}
P
1
=
(
0
,
−
1
,
0
)
and
P
2
=
(
−
1
,
1
,
0
)
\bcb{P_2 = (-1,1,0)}
P
2
=
(
−
1
,
1
,
0
)
Find the point on the line
L
\bcb{L}
L
closest to the point
P
2
\bcb{P_2}
P
2
.
133 - FML 3 - 18.1W e.g. 58.2
Given the vector
v
⃗
=
<
1
,
1
,
2
>
\bcb{\vec{v} = \left< 1, 1, 2 \right>}
v
=
⟨
1
,
1
,
2
⟩
and the points
P
1
=
(
0
,
−
1
,
0
)
\bcb{P_1 = (0,-1,0)}
P
1
=
(
0
,
−
1
,
0
)
and
P
2
=
(
−
1
,
1
,
0
)
\bcb{P_2 = (-1,1,0)}
P
2
=
(
−
1
,
1
,
0
)
Find the
minimum
distance of
P
2
\bcb{P_2}
P
2
to the line
L
\bcb{L}
L
found above.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 56
Given the points
P
1
=
(
1
,
0
,
5
)
\bcb{P_1 = (1, 0, 5)}
P
1
=
(
1
,
0
,
5
)
and
P
2
=
(
3
,
1
,
−
2
)
\bcb{P_2 = (3, 1, -2)}
P
2
=
(
3
,
1
,
−
2
)
, which of the following points are on the line connecting
P
1
\bcb{P_1}
P
1
and
P
2
\bcb{P_2}
P
2
?
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 41
Let
P
=
(
−
3
,
−
3
,
−
4
)
\bcb{P = (-3, -3, -4)}
P
=
(
−
3
,
−
3
,
−
4
)
,
Q
=
(
−
5
,
−
6
,
−
3
)
\bcb{Q = (-5,-6,-3)}
Q
=
(
−
5
,
−
6
,
−
3
)
,
R
=
(
−
6
,
−
4
,
−
6
)
\bcb{R = (-6,-4,-6)}
R
=
(
−
6
,
−
4
,
−
6
)
and
S
=
(
−
2
,
−
3
,
−
4
)
\bcb{S = (-2, -3, -4)}
S
=
(
−
2
,
−
3
,
−
4
)
. Let
l
1
\bcb{l_1}
l
1
be the line passing through
P
\bcb{P}
P
and
Q
\bcb{Q}
Q
, and let
l
2
\bcb{l_2}
l
2
be the line passing through
R
\bcb{R}
R
and
S
\bcb{S}
S
. Find the distance between
R
\bcb{R}
R
and
l
1
\bcb{l_1}
l
1
.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 58.1
Given the vector
v
⃗
=
<
1
,
1
,
2
>
\bcb{\vec{v} = \left< 1, 1, 2 \right>}
v
=
⟨
1
,
1
,
2
⟩
and the points
P
1
=
(
0
,
−
1
,
0
)
\bcb{P_1 = (0,-1,0)}
P
1
=
(
0
,
−
1
,
0
)
and
P
2
=
(
−
1
,
1
,
0
)
\bcb{P_2 = (-1,1,0)}
P
2
=
(
−
1
,
1
,
0
)
, find the scalar-parametric equation of the line
L
\bcb{L}
L
parallel to
v
⃗
\bcb{\vec{v}}
v
that passes through
P
1
\bcb{P_1}
P
1
.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 58.2
Given the vector
v
⃗
=
<
1
,
1
,
2
>
\bcb{\vec{v} = \left< 1, 1, 2 \right>}
v
=
⟨
1
,
1
,
2
⟩
and the points
P
1
=
(
0
,
−
1
,
0
)
\bcb{P_1 = (0,-1,0)}
P
1
=
(
0
,
−
1
,
0
)
and
P
2
=
(
−
1
,
1
,
0
)
\bcb{P_2 = (-1,1,0)}
P
2
=
(
−
1
,
1
,
0
)
, find the
minimum
distance of
P
2
\bcb{P_2}
P
2
to the line
L
\bcb{L}
L
that is parallel to
v
⃗
\bcb{\vec{v}}
v
that passes through
P
1
\bcb{P_1}
P
1
.
133 - FML 3 - 18.1W - e.g. 8
If a line is given by
y
=
m
x
\bcb{\boldsymbol{ y = mx}}
y
=
mx
, find a unit vector pointing in the direction of the line (
i.e.
find the directional vector).