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NDA Math Vector Algebra
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© examsnet.com
Question : 85
Total: 101
Let α,β,γ be distinct real numbers.These points with position vectors
α
∧
i
+
β
∧
j
+
γ
∧
k
,
β
∧
i
+
γ
∧
j
+
α
∧
k
and
γ
∧
i
+
α
∧
j
+
β
∧
k
[NDA I 2015]
are collinear
form an equilateral triangle
form a scalene triangle
form a right-angled triangle
Validate
Solution:
Consider the given points as the vertices of a triangle.
A
=
(
α
,
β
,
γ
)
B
=
(
β
,
γ
,
α
)
C
=
(
γ
.
α
,
β
)
Hence the following sides will be the sides of the triangle
→
A
B
=
→
a
=
(
β
−
α
)
∧
i
+
(
γ
−
β
)
∧
j
+
(
α
−
γ
)
∧
k
→
B
C
=
→
b
=
(
γ
−
β
)
∧
i
+
(
α
−
γ
)
∧
j
+
(
β
−
α
)
∧
k
→
C
A
=
→
c
=
(
α
−
γ
)
∧
i
+
(
β
−
α
)
∧
j
+
(
γ
−
β
)
∧
k
Therefore
a
=
√
(
β
−
α
)
2
+
(
γ
−
β
)
2
+
(
α
−
γ
)
2
b
=
√
(
β
−
α
)
2
+
(
γ
−
β
)
2
+
(
α
−
γ
)
2
c
=
√
(
β
−
α
)
2
+
(
γ
−
β
)
2
+
(
α
−
γ
)
2
∴
a
=
b
=
c
Hence all the sides of the triangle thus formed are equal.
Therefore the triangle is an equilateral triangle.
© examsnet.com
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