KVPY SX SB Exam 2013 Question Paper

Question : 17 of 120

Marks: +1, -0

Let

v_1↖{→}, v_2↖{→}, v_3↖{→}, v_4↖{→}

be unit vectors in the xy -plane, one each in the interior of the four quadrants. Which of the following statements is necessarily true?

$v_1↖{→}+ v_2↖{→}+ v_3↖{→}+v_4↖{→}=0$
There exist i, j with $1 ≤ i < j ≤ 4$ such that $v_i↖{→}+v_j↖{→}$ is in the first quadrant
There exist i, j with $1 ≤ i < j ≤ 4$ such that $v_i↖{→}.v_j↖{→} < 0$
There exist i, j with $1 ≤ i < j ≤ 4$ such that $v_i↖{→}.v_j↖{→} > 0$

Solution:

We have,

v_{1}, v_{2}, v_{3}, v_{4}

be unit vectors lie in

X Y

-plane. Let

v_{1}=\cos α i+\sin α j \;\; α ∈(0,90^{∘})

v_{2}=\cos β i+\sin β j \;\; β ∈(90^{∘}, 180^{∘})

v_{3}=\cos γ i+\sin γ j\;\; γ ∈(180^{∘}, 270^{∘})

v_{1}=\cos δ i+\sin δ j \;\; δ ∈(270^{∘}, 360^{∘})

(a)

v_{1}+v_{2}+v_{3}+v_{4}=0

not necessarily true for all

v_{1}, v_{2}, v_{3}

and

v_{4}

(b) $v_{i}+v_{j}(1 ≤ i

v_{1}+v_{2}=(\cos α+\cos β) i+(\sin α+\sin β) j

y

coordinate is positive i.e.

α

in 1 st quadrant

β

in 2 nd quadrant.
But in this case

x

-coordinate is not necessarily positive.

(c) $ \;\; v_{i}=v_{j}(1 ≤ i Let

α

in 1 st quadrant,

γ

in 3 rd quadrant

∴ \;\; v_{i} ⋅ v_{j}<0

(d)

v_{i} ⋅ v_{j}=(\cos α \cos β)+\sin α \sin β

\;\; =\cos (α-β) \;\text " is positive "\; 0<|α-β|<\;{π}/{2}

\;\; =\;{3 π}/{2}<|α-β|<2 π

Not positive for all values of

v_{1}, v_{2}, v_{3}, v_{4}

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