Derivation of cartesian form from vector form
Let the coordinates of the given point A be (x1, y1, z1) and the direction  ratios of
the line be a, b, c Consider the coordinates of any point P be (x, y, z) Then
zk
jy
ix
r
ˆ
ˆ
ˆ
+
+
r=
; 
k
z
j
y
xi
a
ˆ
ˆ
ˆ
1
1
1
+
+
=
r
and
ˆ
ˆ
ˆ
b
a i
b j
c k
=
+
+
r
Substituting these values in (1) and equating the coefficients of ˆ
ˆ
,i
j  and kˆ , we get
x = x1 + λa;  y = y1 + λ b;  z = z1+ λc (2)
Fig 11