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Byju's Answer
Standard XII
Mathematics
Bisector of Angle between Two Vectors
Show that the...
Question
Show that the points A (−1, 4, −3), B (3, 2, −5), C (−3, 8, −5) and D (−3, 2, 1) are coplanar.
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Solution
The
points
A
,
B
,
C
and
D
will
be
coplanar
iff
any
one
of
the
following
triads
of
vectors
are
coplanar
:
A
B
→
,
A
C
→
,
A
D
→
;
A
B
→
,
B
C
→
,
C
D
→
;
B
C
→
,
B
A
→
,
B
D
,
→
etc
.
T
o
show
that
A
B
→
,
A
C
→
,
A
D
→
are
coplanar
,
we
have
to
prove
that
their
scaler
triple
product
,
i
.
e
.
A
B
→
A
C
→
A
D
→
=
0
Now
,
A
B
→
=
3
-
-
1
i
^
+
2
-
4
j
^
+
-
5
-
-
3
k
^
=
4
i
^
-
2
j
^
-
2
k
^
A
C
→
=
-
3
-
-
1
i
^
+
8
-
4
j
^
+
-
5
-
-
3
k
^
=
-
2
i
^
+
4
j
^
-
2
k
^
A
D
→
=
-
3
-
-
1
i
^
+
2
-
4
j
^
+
1
-
-
3
k
^
=
-
2
i
^
-
2
j
^
+
4
k
^
∴
A
B
→
A
C
→
A
D
→
=
4
-
2
-
2
-
2
4
-
2
-
2
-
2
4
=
4
16
-
4
+
2
-
8
-
4
-
2
4
+
8
=
0
Thus
,
the
given
points
are
coplanar
.
Suggest Corrections
0
Similar questions
Q.
Examine if the following set of points are coplanar: If they are coplanar write 1 otherwise write 0.
(
3
,
2
,
−
5
)
,
(
−
3
,
8
,
−
5
)
(
−
3
,
2
,
1
)
,
(
−
1
,
4
,
−
3
)
.
Q.
Show that
D
(
−
1
,
4
,
−
3
)
is the centroid of
Δ
A
B
C
with the vertices
A
(
3
,
2
,
−
5
)
,
B
(
−
3
,
8
,
−
5
)
and
C
(
−
3
,
2
,
1
)
.
Q.
Show that
(
−
1
,
4
,
−
3
)
is the circum-centre of the triangle formed by the points
(
3
,
2
,
−
5
)
,
(
−
3
,
8
,
−
5
)
and
(
−
3
,
2
,
1
)
.
Q.
Find the value of
x
such that the points
A
(
3
,
2
,
1
)
,
B
(
4
,
x
,
5
)
,
C
(
4
,
2
,
−
2
)
and
D
(
6
,
5
,
−
1
)
are coplanar.
Q.
Find λ for which the points A (3, 2, 1), B (4, λ, 5), C (4, 2, −2) and D (6, 5, −1) are coplanar.
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