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Byju's Answer
Standard XII
Mathematics
Bisector of Angle between Two Vectors
Show that fou...
Question
Show that four points whose position vectors are
6
i
^
-
7
j
^
,
16
i
^
-
19
j
^
-
4
k
^
,
3
i
^
-
6
k
^
,
2
i
^
-
5
j
^
+
10
k
^
are coplanar.
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Solution
DISCLAIMER: Given points are not coplanar.
Let
A
,
B
,
C
,
D
be
the
given
points
.
The
given
points
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
.
In
order
to
show
that
A
B
→
,
A
C
→
,
A
D
→
are
coplanar
,
we
will
have
to
show
that
their
scaler
triple
product
i
.
e
.
A
B
→
A
C
→
A
D
→
=
0
Using
,
P
Q
→
=
Position
vector
of
Q
-
Position
vector
of
P
,
we
obtain
Now
,
A
B
→
=
16
i
^
-
19
j
^
-
4
k
^
-
6
i
^
-
7
j
^
=
10
i
^
-
12
j
^
-
4
k
^
A
C
→
=
3
i
^
-
6
k
^
-
6
i
^
-
7
j
^
=
-
3
i
^
+
7
j
^
-
6
k
^
and
,
A
D
→
=
2
i
^
-
5
j
^
+
10
k
^
-
6
i
^
-
7
j
^
=
-
4
i
^
+
2
j
^
+
10
k
^
∴
A
B
→
A
C
→
A
D
→
=
10
-
12
-
4
-
3
7
-
6
-
4
2
10
=
10
70
+
12
+
12
-
30
-
24
-
4
-
6
+
28
=
84
Thus
,
the
given
points
are
not
coplanar
.
Suggest Corrections
0
Similar questions
Q.
Show that the four points having position vectors
6
i
^
-
7
j
^
,
16
i
^
-
19
j
^
-
4
k
^
,
3
j
^
-
6
k
^
,
2
i
^
-
5
j
^
+
10
k
^
are coplanar.
Q.
If the points whose position vectors are
3
¯
i
−
2
¯
j
−
¯
¯
¯
k
,
2
¯
i
+
3
¯
j
−
4
¯
¯
¯
k
,
−
¯
i
+
¯
j
+
2
¯
¯
¯
k
and
4
¯
i
+
5
¯
j
+
λ
¯
¯
¯
k
are coplanar then show that
λ
=
−
146
m
.Find
m
Q.
Find the value of λ for which the four points with position vectors
-
j
^
-
k
^
,
4
i
^
+
5
j
^
+
λ
k
^
,
3
i
^
+
9
j
^
+
4
k
^
and
-
4
i
^
+
4
j
^
+
4
k
^
are coplanar.
Q.
Prove that the position vectors
4
i
+
5
j
+
6
k
,
5
i
+
6
j
+
4
k
,
&
6
i
+
4
j
+
5
k
form an equilateral triangle.
Q.
Find the vector equation of the plane passing through points
4
i
−
3
j
−
k
,
3
i
+
7
j
−
10
k
and
2
i
+
5
j
−
7
k
and show that the point
i
+
2
j
−
3
k
lies in the plane.
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