NIMCET 2007 Mathematics PYQ — Let be distinct real numbers. The points with position vectors , … | Mathem Solvex | Mathem Solvex
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NIMCET 2007 — Mathematics PYQ
NIMCET | Mathematics | 2007
Let α,β,γ be distinct real numbers. The points with position vectors αi^+βj^+γk^, βi^+γj^+αk^, and γi^+αj^+βk^:
Choose the correct answer:
A.
are collinear
B.
form an equilateral triangle
(Correct Answer)
C.
form a scalene triangle
D.
form a right angled triangle
Correct Answer:
form an equilateral triangle
Explanation
Let the three given points be represented as A, B, and C. Their position vectors are:
a=αi^+βj^+γk^
b=βi^+γj^+αk^
c=γi^+αj^+βk^
To determine the nature of the triangle formed by these points, we need to find the lengths of the sides AB, BC, and CA using the distance formula or vector magnitudes.
Step 1: Find the side length AB
The vector AB is given by:
AB=b−a=(β−α)i^+(γ−β)j^+(α−γ)k^
The magnitude (length) of AB is:
AB=∣AB∣=(β−α)2+(γ−β)2+(α−γ)2
Step 2: Find the side length BC
The vector BC is given by:
BC=c−b=(γ−β)i^+(α−γ)j^+(β−α)k^
The magnitude (length) of BC is:
BC=∣BC∣=(γ−β)2+(α−γ)2+(β−α)2
Step 3: Find the side length CA
The vector CA is given by:
CA=a−c=(α−γ)i^+(β−α)j^+(γ−β)k^
The magnitude (length) of CA is:
CA=∣CA∣=(α−γ)2+(β−α)2+(γ−β)2
Conclusion
Observing the expressions for the lengths of the three sides:
AB=BC=CA=(α−β)2+(β−γ)2+(γ−α)2
Since all three sides are equal in length, the points A, B, and C form an equilateral triangle.
Furthermore, because α,β,γ are distinct real numbers, the side length is strictly greater than zero (AB=0), ensuring that the points do not coincide or become collinear.
Correct Option: (B) Form an equilateral triangle
Explanation
Let the three given points be represented as A, B, and C. Their position vectors are:
a=αi^+βj^+γk^
b=βi^+γj^+αk^
c=γi^+αj^+βk^
To determine the nature of the triangle formed by these points, we need to find the lengths of the sides AB, BC, and CA using the distance formula or vector magnitudes.
Step 1: Find the side length AB
The vector AB is given by:
AB=b−a=(β−α)i^+(γ−β)j^+(α−γ)k^
The magnitude (length) of AB is:
AB=∣AB∣=(β−α)2+(γ−β)2+(α−γ)2
Step 2: Find the side length BC
The vector BC is given by:
BC=c−b=(γ−β)i^+(α−γ)j^+(β−α)k^
The magnitude (length) of BC is:
BC=∣BC∣=(γ−β)2+(α−γ)2+(β−α)2
Step 3: Find the side length CA
The vector CA is given by:
CA=a−c=(α−γ)i^+(β−α)j^+(γ−β)k^
The magnitude (length) of CA is:
CA=∣CA∣=(α−γ)2+(β−α)2+(γ−β)2
Conclusion
Observing the expressions for the lengths of the three sides:
AB=BC=CA=(α−β)2+(β−γ)2+(γ−α)2
Since all three sides are equal in length, the points A, B, and C form an equilateral triangle.
Furthermore, because α,β,γ are distinct real numbers, the side length is strictly greater than zero (AB=0), ensuring that the points do not coincide or become collinear.
