Question

An arc AC of a circle subtends a right angle at the center O. The point B divides the arc in the ratio 1:2. If
\[\overrightarrow {OA} = \overrightarrow a\] and \[\overrightarrow {OB} = \overrightarrow b\], then the vector \[\overrightarrow {OC}\] in terms of \[\overrightarrow a \& \overrightarrow b\], is
A. \[\sqrt 3 \overrightarrow a - 2\overrightarrow b\]
B. \[- \sqrt 3 \overrightarrow a + 2\overrightarrow b\]
C. \[2\overrightarrow a - \sqrt 3 \overrightarrow b\]
D. \[- 2\overrightarrow a + \sqrt 3 \overrightarrow b\]

Answer 1

Hint: In this problem, we need to apply the vector law of addition in triangle OBD to obtain the vector OC. The vector law of addition says that, if two vectors are represented as two sides of the triangle having order of direction and magnitude, then the third side of the triangle represents the resultant of the two vectors.

Complete step by step solution:
Given:
An arc AC of a circle subtends a right angle at the center O and, the point B divides the arc in the ratio 1:2.

Since, the point B divides the arc AC in a ratio of 1:2, it can be written as follows:
\[\begin{gathered}
  \overrightarrow {OD} = \dfrac{{\overrightarrow {OC} }}{2} \\
  \overrightarrow {OE} = \dfrac{{\sqrt 3 }}{2}\overrightarrow {OA} \\
\end{gathered}\]
Apply the vector law of addition in the triangle \[OBD\] as shown below.

\[
  \,\,\,\,\,\,\overrightarrow {OD} + \overrightarrow {DB} = \overrightarrow {OB} \\
   \Rightarrow \overrightarrow {OD} + \overrightarrow {OE} = \overrightarrow {OB} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\overrightarrow {OE} = \overrightarrow {DB} } \right) \\
   \Rightarrow \overrightarrow {OB} = \dfrac{{\overrightarrow {OC} }}{2} + \dfrac{{\sqrt 3 }}{2}\overrightarrow {OA} \\
   \Rightarrow 2\overrightarrow {OB} = \overrightarrow {OC} + \sqrt 3 \overrightarrow {OA} \\
   \Rightarrow 2\overrightarrow b = \overrightarrow {OC} + \sqrt 3 \overrightarrow a \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\overrightarrow {OA} = \overrightarrow a \,\,{\text{and}}\,\,\overrightarrow {OB} = \overrightarrow b } \right) \\
   \Rightarrow \overrightarrow {OC} = 2\overrightarrow b - \sqrt 3 \overrightarrow a \\
\]

Thus, the vector \[\overrightarrow {OC} \] is \[2\overrightarrow b - \sqrt 3 \overrightarrow a\], hence, option (B) is the correct answer.

Note: Two vectors can replace each other if they are equal in magnitude and direction. The sum of the two vectors having equal in magnitude and opposite in direction is a null vector.