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Byju's Answer
Standard XII
Mathematics
Addition of Vectors
P is point in...
Question
P is point in the plane of the triangle ABC. pv's of A,B and c.
→
a
,
→
b
and
→
c
respectively with respect to P as the origin.
If
(
→
b
+
→
c
)
⋅
(
→
b
−
→
c
)
=
0
and
(
→
c
+
→
a
)
⋅
(
→
c
−
→
a
)
=
0
then w.r.t. the triangle ABC,P is its
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Solution
(
→
b
+
→
c
)
⋅
(
→
b
−
→
c
)
=
0
∣
∣
→
b
∣
∣
2
−
|
c
|
2
=
0
|
→
a
|
=
∣
∣
→
b
∣
∣
=
|
→
c
|
(
→
c
+
→
a
)
⋅
(
→
c
−
→
a
)
=
0
|
→
c
|
2
−
|
→
a
|
2
=
0
|
c
|
=
|
a
|
Implies P is cir-cum center of triangle
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Similar questions
Q.
P) P is a point in the plane of the triangle ABC. Pv's of A,B and C are
→
a
,
→
b
a
n
d
→
c
respectively with respect to P as the origin. If
(
→
b
+
→
c
)
.
(
→
b
−
→
c
)
=
0
a
n
d
(
→
c
+
→
a
)
.
(
→
c
−
→
a
)
=
0
then w.r.t. the Triangle ABC, P is its
1)Centroid
Q) If
→
a
,
→
b
,
→
c
a
r
e
the position vectors of the three non collinear points A,B and C repectively such that the vector
→
V
=
−
−
→
P
A
+
−
−
→
P
B
+
−
−
→
P
C
is a null vector then w.r.t. the
Δ
A
B
C
, P is its
2)orthocentre
R)if P is a point inside the
Δ
A
B
C
such that the vector
→
R
=
(
B
C
)
(
−
−
→
P
A
)
+
C
A
(
−
−
→
P
B
)
+
(
A
B
)
(
−
−
→
P
C
)
is null vector then w.r.t. the
Δ
A
B
C
, P is its
3)Incentre
S) If P is a point in the plane of the triangle ABC such that the scalar product
−
−
→
P
A
.
−
−
→
C
B
a
n
d
−
−
→
P
B
.
−
−
→
A
C
vanishes, then w.r.t. the
Δ
A
B
C
. P is its
4)Circumcentre
Q.
A
,
B
,
C
,
D
are four points on a plane with position vectors
a
,
b
,
c
,
d
,
respectively, such that
(
a
−
d
)
.
(
b
−
c
)
=
(
b
−
d
)
.
(
c
−
a
)
=
0.
For
△
A
B
C
,
D
is the
Q.
A
,
B
,
C
and
D
are four points in a plane with Position Vectors
→
a
,
→
b
,
→
c
and
→
d
respectively such that
(
→
a
−
→
d
)
.
(
→
b
−
→
c
)
=
(
→
b
−
→
d
)
.
(
→
c
−
→
a
)
=
0
,
then for the triangle
A
B
C
,
D
is its
Q.
In a
△
A
B
C
, a,b and c are the side of the triangle opposite to the angles
A
,
B
,
C
respectively. Then the value of
a
3
sin
(
B
−
C
)
+
b
3
sin
(
C
−
A
)
+
c
3
sin
(
A
−
B
)
is equal to:
Q.
A point
P
moves on a plane
x
a
+
y
b
+
z
c
=
1
and
O
be the origin. A plane through
P
and perpendicular to
O
P
meets the coordinate axes at
A
,
B
and
C
respectively. Let
G
be the centroid of the triangle
A
B
C
. If the planes through
A
,
B
and
C
parallel to the planes
x
=
0
,
y
=
0
and
z
=
0
, respectively, intersect at
Q
, then
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