NIMCET 2010 Mathematics PYQ — Let and be three non zero vectors, no two of which are collinear … | Mathem Solvex | Mathem Solvex
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NIMCET 2010 — Mathematics PYQ
NIMCET | Mathematics | 2010
Let a,b and c be three non zero vectors, no two of which are collinear and the vector a+b is collinear with c while b+c is collinear with a, then a+b+c is equal to:
Choose the correct answer:
A.
a
B.
b
C.
c
D.
None
(Correct Answer)
Correct Answer:
None
Explanation
Step 1: Express the collinear conditions using scalar multiples.
Two vectors are collinear if one can be expressed as a scalar multiple of the other.
Since a+b is collinear with c, there exists a scalar λ such that:
a+b=λc…(Equation 1)
Since b+c is collinear with a, there exists a scalar μ such that:
b+c=μa…(Equation 2)
Step 2: Eliminate b from the equations.
From Equation 1, we have b=λc−a.
Substitute this value of b into Equation 2:
(λc−a)+c=μa
λc+c−a=μa
(λ+1)c=(μ+1)a
Step 3: Analyze the relationship between a and c.
The problem states that no two vectors are collinear. This means a and c are not collinear.
For the equation (λ+1)c=(μ+1)a to hold true when the vectors themselves are not collinear, the scalar coefficients on both sides must be zero.
λ+1=0⟹λ=−1
μ+1=0⟹μ=−1
Step 4: Find the value of a+b+c.
Substitute λ=−1 back into Equation 1:
a+b=−1(c)
a+b=−c
Adding c to both sides:
a+b+c=0
Step 5: Compare with options.
The result is the null vector (0), which is not listed in options (a), (b), or (c).
Correct Option: (d) None
Explanation
Step 1: Express the collinear conditions using scalar multiples.
Two vectors are collinear if one can be expressed as a scalar multiple of the other.
Since a+b is collinear with c, there exists a scalar λ such that:
a+b=λc…(Equation 1)
Since b+c is collinear with a, there exists a scalar μ such that:
b+c=μa…(Equation 2)
Step 2: Eliminate b from the equations.
From Equation 1, we have b=λc−a.
Substitute this value of b into Equation 2:
(λc−a)+c=μa
λc+c−a=μa
(λ+1)c=(μ+1)a
Step 3: Analyze the relationship between a and c.
The problem states that no two vectors are collinear. This means a and c are not collinear.
For the equation (λ+1)c=(μ+1)a to hold true when the vectors themselves are not collinear, the scalar coefficients on both sides must be zero.
λ+1=0⟹λ=−1
μ+1=0⟹μ=−1
Step 4: Find the value of a+b+c.
Substitute λ=−1 back into Equation 1:
a+b=−1(c)
a+b=−c
Adding c to both sides:
a+b+c=0
Step 5: Compare with options.
The result is the null vector (0), which is not listed in options (a), (b), or (c).
