Chapter 11 Three Dimensional Geometry Worksheet Page 6
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MATHEMATICS
11.3 Equation of a Line in Space
We have studied equation of lines in two dimensions in Class XI, we shall now study
the vector and cartesian equations of a line in space.
A line is uniquely determined if
(i) it passes through a given point and has given direction, or
(ii) it passes through two given points.
H
11.3.1 Equation of a line through a given point and parallel to a given vector
b
H
Let
be the position vector of the given point
a
A with respect to the origin O of the
rectangular coordinate system. Let l be the
line which passes through the point A and is
H
H
parallel to a given vector
. Let
be the
b
r
position vector of an arbitrary point P on the
line (Fig 11.4).
H
KKK H
Then
is parallel to the vector
b
, i.e.,
AP
H
KKK H
= λ
, where λ is some real number.
Fig 11.4
b
AP
KKK H
KKK H
KKKH
AP
OP – OA
=
But
H
H
H
−
λ
r
a
i.e.
b
=
Conversely, for each value of the parameter λ, this equation gives the position
vector of a point P on the line. Hence, the vector equation of the line is given by
H
H
H
ë
r
a +
b
=
... (1)
H
=
+
+
ˆ
ˆ
ˆ
Remark
If
, then a, b, c are direction ratios of the line and conversely,
b
ai bj ck
H
=
+
+
ˆ
ˆ
ˆ
if a, b, c are direction ratios of a line, then
will be the parallel to
b
ai bj ck
H
the line. Here, b should not be confused with |
b
|.
Derivation of cartesian form from vector form
Let the coordinates of the given point A be (x
, y
, z
) and the direction ratios of
1
1
1
the line be a, b, c. Consider the coordinates of any point P be (x, y, z). Then
H
H
=
+
+
ˆ
=
+
+
ˆ
ˆ
ˆ
ˆ
ˆ
;
r
i x
j y
k z
a
x
i
y
j
z
k
H
1
1
1
=
+
+
ˆ
ˆ
ˆ
and
b
a i
b j
c k
ˆ ˆ
k ˆ
, i j
Substituting these values in (1) and equating the coefficients of
and
, we get
+ λ a; y = y
+ λ b; z = z
+ λ c
x = x
... (2)
1
1
1
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