The length of a side of a square OPQR is \(\mathrm{a}, \mathrm{O}\) is the origin OP and \(\mathrm{OR}\) are along positive direction of the \(\mathrm{X}\) and \(\mathrm{Y}\) axes respectively. If \(\mathrm{A}\) and \(\mathrm{B}\) are mid points of \(\underline{\mathrm{PQ}}\) and \(\underline{\mathrm{QR}}\) respectively then measure of angle between \(\underline{\mathrm{OA}}\) and \(\underline{\mathrm{OB}}\) is.... (a) \(\cos ^{-1}(3 / 5)\) (b) \(\tan ^{-1}(4 / 3)\) (c) \(\cot ^{-1}(3 / 4)\) (d) \(\sin ^{-1}(3 / 5)\)

In solving geometry problems on the JEE, one must be well-versed in coordinate geometry. This branch of mathematics allows us to analyze figures like squares, triangles, and circles through the lens of algebra—using coordinates and equations.

For instance, in the problem about the square OPQR, we identified the coordinates of points P and Q and then used the midpoint theorem to find the coordinates of points A and B. The theorem states that the midpoint's coordinates are the averages of the coordinates of the endpoints of the segment. So correctly understanding and applying the midpoint formula is crucial in problems that require finding distance or midpoint between two points on a coordinate plane.

Furthermore, understanding the Cartesian plane layout is essential as it aids visualization. The X and Y axes themselves form a square or rectangle's sides in the plane, making it easier to interpret the problem and solve it efficiently.

With vector algebra, we move from static points to dynamic quantities that have both magnitude and direction. In our textbook problem, we used vectors to represent the positions of points A and B relative to the origin O.

The position vector of a point is simply a vector that extends from the origin to that point. It is characterized by its components along the X and Y axes in 2D problems. Calculating these position vectors is fundamental in problems involving movement or direction, such as those posed in the JEE exams.

In vector algebra, it's also critical to understand operations like vector addition, scaling, and most pertinently for this problem, the dot product—which is used to calculate the angle between vectors. Always remember, the notation \(\begin{bmatrix} x \ y \ end{bmatrix}\) represents a vector where 'x' and 'y' are components along the X and Y axes respectively.

The dot product (also known as scalar product) is an operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This operation is central to problems that involve angles or projections.

In this exercise, we found the dot product of vectors \(\begin{bmatrix} a \ \frac{a}{2} \ end{bmatrix}\) and \(\begin{bmatrix} \frac{a}{2} \ a \ end{bmatrix}\), which allowed us to establish a relationship between the length of the vectors and the cosine of the angle between them. Following this approach, the student should remember the dot product formula: \(\begin{bmatrix} x_1 \ y_1 \end{bmatrix}\cdot \begin{bmatrix} x_2 \ y_2 \end{bmatrix} = x_1x_2 + y_1y_2\), which in our case simplifies to finding the cosine of the angle being searched for.

The dot product concept is not only useful in geometry, but it's also prevalent in other fields, like physics and engineering, where it's used to calculate work done or to determine orthogonality in vector spaces.