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Byju's Answer
Standard XII
Mathematics
Dot Product
Show that the...
Question
Show that the points
O
(
0
,
0
)
,
A
(
2
,
−
3
,
3
)
,
B
(
−
2
,
3
,
−
3
)
are collinear. Find the ratio in which each point divides the segment joining the other two.
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Solution
Let's find area of
△
O
A
B
which is given by,
Δ
=
1
2
|
O
A
|
|
O
B
|
sin
θ
where
θ
is angle between
→
O
A
and
→
O
B
Now,
cos
θ
=
→
O
A
.
→
O
A
∣
∣
→
O
A
∣
∣
∣
∣
→
O
B
∣
∣
=
−
4
−
9
−
9
√
4
+
9
+
9
√
4
+
9
+
9
=
−
22
22
⇒
cos
θ
=
−
1
⇒
θ
=
180
o
⇒
Δ
=
0
as
sin
180
=
0
⇒
O
,
A
,
B
are collinear
Now, let
A
divides
→
O
B
in
k
:
1
⇒
2
=
−
2
k
+
0
k
+
1
⇒
2
k
+
2
=
−
2
k
⇒
4
k
=
−
2
⇒
k
=
−
1
2
⇒
A
divides
→
O
B
in
1
:
2
externally
Now for
B
⇒
−
2
=
2
k
+
0
k
+
1
⇒
−
2
k
−
2
=
2
k
⇒
4
k
=
−
2
⇒
k
=
−
1
2
⇒
B
divides
→
O
A
in
1
:
2
externally for
O
⇒
O
=
−
2
k
+
2
k
+
1
⇒
2
k
=
2
⇒
k
=
1
⇒
O
divides
→
A
B
in
1
:
1
internally
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