Therefore,
P is on the line if and only if
(
)
r
a
b
a
−=λ −
r
r
r
r
Fig 11 5
© NCERT
not to be republished
 MATHEMATI CS
470
or
(
)
r
a
b
a
=
+ λ
−
r
r
r
r , λ ∈ R (1)
This is the vector equation of the line Derivation of cartesian form from vector form
We have
1
1
1
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
,
r
xi
y j
z k a
x i
y j
z k
= + +
= + +
r
r
and 
2
2
2 ˆ
ˆ
ˆ
,
b
x i
y j
z k
= + +
r
Substituting these values in (1), we get
$
$
$
$
$
$
1
1
1
2
1
2
1
2
1
[(
)
(
)
(
) ]
xi
y j
z k
x i
y j
z k
x
x i
y
y
j
z
z k
+
+
=
+
+
+ λ
−
+
−
+
−
$
$
$
Equating the like coefficients of 
jk
i
ˆ
,ˆ
,ˆ
, we get
x = x1 + λ (x2 – x1); y = y1 + λ (y2 – y1); z = z1 + λ (z2 – z1)
On eliminating  λ, we obtain
1
1
1
2
1
2
1
2
1
x
x
y
y
z
z
x
x
y
y
z
z
−
−
−
=
=
−
−
−
which is the equation of the line in Cartesian form