Consider a square OPQR having the length of side a, where \(\mathrm{O}(0,0)\). The sides \(\underline{\mathrm{OP}}\) and \(\underline{\mathrm{OR}}\) are along the positive \(\mathrm{X}\) -axis and Y-axis respectively. If \(\mathrm{A}\) and \(\mathrm{B}\) are the mid points of \(\underline{\mathrm{PQ}}\) and \(Q \underline{R}\) respectively, then the angle between \(\underline{\mathrm{OA}}\) and \(\underline{\mathrm{OB}}\) would be \(\ldots \ldots \ldots\) (a) \(\cos ^{-1}(3 / 5)\) (b) \(\tan ^{-1}(4 / 3)\) (c) \(\cos ^{-1}(3 / 4)\) (d) \(\sin ^{-1}(3 / 5)\)

: As point O is at the origin, and sides OP and OR are along the positive X-axis and Y-axis, the coordinates of points P and R can be determined directly: - Point P: (a, 0) - Point R: (0, a) To find the coordinates of point Q, we can either add the coordinates of P and R or remember that it's a square and the point Q will have equal distance from X and Y-axes. Hence, - Point Q: (a, a)

: Now, we will find coordinates of midpoints A and B by averaging the coordinates of the endpoints of the respective line segments. - Midpoint formula: \(\text{M}=\left(\frac{X_1 + X_2}{2},\frac{Y_1 + Y_2}{2}\right)\) Calculate Midpoint A, which is the midpoint of line segment PQ: \(A = \left(\frac{a + a}{2},\frac{0 + a}{2}\right) = \left(\frac{2a}{2},\frac{a}{2}\right) = (a, \frac{a}{2})\) Calculate Midpoint B, which is the midpoint of line segment QR: \(B = \left(\frac{a + 0}{2},\frac{a + a}{2}\right) = \left(\frac{a}{2},\frac{2a}{2}\right) = \left(\frac{a}{2}, a\right)\)

: Now, we will find vectors OA and OB by subtracting the coordinates of the origin from the coordinates of points A and B respectively. - Vector OA: \(\mathrm{A} - \mathrm{O} = (a, \frac{a}{2}) - (0, 0) = (a, \frac{a}{2})\) - Vector OB: \(\mathrm{B} - \mathrm{O} = \left(\frac{a}{2}, a\right) - (0, 0) = \left(\frac{a}{2}, a\right)\)

: The angle between two vectors can be calculated using the dot product formula as follows: - Dot product formula: \(\vec{u}\cdot\vec{v} = ||\vec{u}|| \cdot ||\vec{v}|| \cdot \cos{\theta}\) - The angle between vectors \(\vec{u}\) and \(\vec{v}\) is given by \(\theta = \cos^{-1}{\frac{\vec{u}\cdot\vec{v}}{||\vec{u}|| \cdot ||\vec{v}||}}\) We'll follow these steps: 1. Calculate the dot product of vectors OA and OB: \(\vec{OA}\cdot\vec{OB} = (a, \frac{a}{2})\cdot\left(\frac{a}{2}, a\right) = a\cdot\frac{a}{2} + \frac{a}{2}\cdot a = \frac{3}{4}a^2\) 2. Calculate the magnitude of vectors OA and OB: \(||\mathrm{OA}|| = \sqrt{a^2 + \left(\frac{a}{2}\right)^2} = \frac{\sqrt{5}}{2}a\), \(\, ||\mathrm{OB}|| = \sqrt{\left(\frac{a}{2}\right)^2 + a^2} = \frac{\sqrt{5}}{2}a\) 3. Calculate the angle between vectors OA and OB: \(\theta = \cos^{-1}{\frac{\frac{3}{4}a^2}{\frac{5}{4}a^2}} = \cos^{-1}{\frac{3}{5}}\) The angle between vector OA and vector OB is given by option (a) \(\cos^{-1}(3/5)\).