Theorems
Theorem - 1
The vector equation of a plane passing through the point A(
a
) and parallel to two non-collinear vectors
b
and
c
is [
r
b
c
] = [
a
b
c
]
Proof :
Let
OA
=
a
and
OP
=
r
be any point in the plane.
⇒ The vectors
AP
,
b
,
c
are coplanar.
⇒ [
AP
b
c
] = 0
⇒
AP
. (
b
×
c
) = 0
⇒ (
OP
–
OA
) (
b
×
c
) = 0
⇒ (
r
–
a
) (
b
×
c
) = 0
⇒
r
. (
b
×
c
) =
a
. (
b
×
c
)
⇒ [
r
b
c
] = [
a
b
c
]
Theorem - 2
The vector equation of the plane passing through the points A(
a
), B(
b
) and parallel to the vector
c
is [
r
b
c
] + [
r
c
a
] = [
a
b
c
]
Proof :
Let
OA
=
a
,
OB
=
b
and
OP
=
r
be any point in the plane.
∴ The vectors
AP
,
AB
and
C
are coplanar.
⇒ [
AP
AB
C
] = 0
⇒
AP
. (
AB
×
C
) = 0
⇒ (
OP
–
OA
) ((
OB
–
OA
) ×
C
) = 0
⇒ (
r
–
a
) ((
b
–
a
) ×
c
) = 0
⇒ (
r
–
a
) (
b
×
c
+
c
×
a
) = 0
⇒
r
. (
b
×
c
) +
r
. (
c
×
a
) =
a
. (
b
×
c
) +
a
. (
c
×
a
)
⇒ [
r
b
c
] + [
r
c
a
] = [
a
b
c
] (∵ [
a
c
a
] = 0)
Theorem - 3
The vector equation of a line passing through three non-collinear points A(
a
), B(
b
), C(
c
) is [
r
b
c
] + [
r
c
a
] + [
r
a
b
] = [
a
b
c
]
Proof :
Let
OA
=
a
,
OB
=
b
,
OC
=
c
and
OP
=
r
be any point in the plane.
∴ The points A, B, C and P are coplanar.
⇒
AP
,
AB
, and
AC
are coplanar.
⇒
r
–
a
,
b
–
a
and
c
–
a
are coplanar
⇒ [
r
–
a
b
–
a
c
–
a
] = 0
⇒ [
r
b
–
a
c
–
a
] – [
a
b
–
a
c
–
a
] = 0
⇒ [
r
b
c
–
a
] – [
r
a
c
–
a
] – [
a
b
c
–
a
] + [
a
a
c
–
a
] = 0
⇒ [
r
b
c
] – [
r
b
a
] – [
r
a
c
] – [
r
a
a
] – [
a
b
c
] + [
a
b
a
] + 0 = 0
⇒ [
r
b
c
] + [
r
a
b
] – [
r
c
a
] – 0 – [
a
b
c
] = 0
⇒ [
r
b
c
] + [
r
c
a
] + [
r
a
b
] = [
a
b
c
]