In the cartesian form,
it can be expressed as
2
1
2
1
2
1
1
1
1
2
2
2
0
x
x
y
y
z
z
a
b
c
a
b
c
−
−
− = (4)
Example 21 Show that the lines
+3
1
5
–3
1
5
x
y
z
−
−
=
=
 and  
+1
2
5
–1
2
5
x
y
z
−
−
=
=
 are coplanar Solution Here,x1 = – 3, y1 = 1, z1 = 5, a1 = – 3, b1 = 1, c1 = 5
x2 = – 1, y2 = 2, z2 = 5, a2 = –1, b2 = 2, c2 = 5
Now, consider the determinant
2
1
2
1
2
1
1
1
1
2
2
2
2 1
0
3 1 5
0
1
2
5
x
x
y
y
z
z
a
b
c
a
b
c
−
−
− =−
=
−
Therefore, lines are coplanar 11