Sec. 10.5 Equation of Lines and Planes
In 2D a line is determined by a point and a slope (direction).
In 3D a line L is determined by a point
0
0
0
0
(
,
,
)
P
x
y
z
=
and the direction of L which is given by a
vector v that is parallel to L
.
The vector equation of line L:
0
r
r
tv
=
+
(Putting bounds on t will yield a line segment.)
As t varies the line is traced out by the tip of vector r. (v is the directional vector of the line.)
Let v = < a, b, c >
Æ
tv = < at, bt, ct >. Let
r = < x, y, z > and
0
0
0
0
,
,
r
x
y
z
=<
>
.
Substituting into the
vector equation of the line we can get the component form.
The component form of the vector equation of line
L:
0
0
0
,
,
r
x
at y
bt z
ct
=<
+
+
+
>
Since two vectors are equal if and only if their components are equal, we get the parametric form of a line in
3 space where t is a real number.
The parametric form of line L:
0
0
0
x
x
at
y
y
bt
z
z
ct
=
+
⎧
⎪
=
+
⎨
⎪
=
+
⎩
through
0
0
0
0
(
,
,
)
P
x
y
z
=
and parallel to v = < a, b, c >.
Note: The vector form and the parametric form are not unique!
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 Spring '03
 MECothren
 Linear Algebra, Multivariable Calculus, Vector Space, Slope

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