Let $\vec{v}_{1}, \overrightarrow{v_{2}}, \overrightarrow{v_{3}}, \overrightarrow{v_{4}}$ be unit vectors in the $x y$-plane, one each in the interior of the four quadrants. Which of the following statements is necessarily true?

• Correct answer
  A

  $\overrightarrow{v_{1}}+\overrightarrow{v_{2}}+\overrightarrow{v_{3}}+\overrightarrow{v_{4}}=0$
• B

  There exist $\mathrm{i}, \mathrm{j}$ with $1 \leq \mathrm{i} < \mathrm{j} \leq 4$ such $\overrightarrow{v_{i}}+\overrightarrow{v_{j}}$ is in the first quadrant
• C

  There exist $\mathrm{i}, \mathrm{j}$ with $1 \leq \mathrm{i} < \mathrm{j} \leq 4$ such that $\vec{v}_{\mathrm{i}} \cdot \overrightarrow{v_{\mathrm{j}}} < 0$
• D

  There exist i, $j$ with $1 \leq i < j \leq 4$ such that $\vec{v}_{i} \cdot \vec{v}_{j}>0$

Solution

In this case $B, C, D$ are not possible.

Hence, the correct answer is option: 'A'.