\(\mathrm{A}(-2,5), \mathrm{B}(6,2)\), then \(\underline{\mathrm{AB}}-\underline{\mathrm{AB}}=\ldots \ldots \ldots\) (a) \([(8 t-2,5-3 t) /(t<0)]\) (b) \([(8 t-2,5-3 t) /(0 \leq t \leq 1)]\) (c) \([(8 \mathrm{t}-2,5-3 \mathrm{t}) /\\{\mathrm{t} \in \mathrm{R}-[0,1]\\}]\) (d) \([(8 t-2,5-3 t) /(t>1)]\)

Coordinate geometry, also known as analytic geometry, involves the study of geometry using a coordinate system. This method associates each point in the plane with a pair of numerical coordinates, which are the distances to the point from two fixed perpendicular directed lines, measured in the same unit of length.

When we talk about the Cartesian coordinate system, the pair \( (x, y) \) represents the horizontal position \( x \) and the vertical position \( y\) of a point in the plane. Coordinate geometry is fundamental in many fields of mathematics and science since it seamlessly combines algebra and geometry.

In the context of the given exercise, coordinate geometry allows us to treat points A and B as ordered pairs \( A(-2, 5) \) and \( B(6, 2) \) respectively. The subtraction of these points is related to the concept of vector subtraction, which leads us to the next core concept.

Vectors in mathematics are objects that have both magnitude and direction. They are usually represented as directed line segments, and in coordinate geometry, they can be denoted by coordinates. A two-dimensional vector \( \mathbf{v} \) is expressed in the form \( \mathbf{v} = (v_x, v_y) \), where \( v_x \) and \( v_y \) are the components of the vector along the x and y axes.

Subtracting vectors involves taking the corresponding components of one vector away from those of the other. For example, if we have two vectors \(\mathbf{a} = (a_x, a_y)\) and \mathbf{b} = (b_x, b_y)\, then their subtraction is \( \mathbf{a} - \mathbf{b} = (a_x - b_x, a_y - b_y) \).

In the given problem, \(\overline{\mathrm{AB}}\) is a vector resulting from the subtraction of point A from point B. By applying vector subtraction, we can conclude that subtracting \(\overline{\mathrm{AB}}\) from itself gives us \mathbf{0} = (0, 0)\, the zero vector, which has no magnitude or direction.

The Joint Entrance Examination (JEE) is an engineering entrance assessment conducted in India and is known for its challenging and complex math problems. JEE Maths problems often delve into advanced algebra, calculus, and geometry concepts, requiring a deep understanding and the ability to solve multifaceted and multi-step problems.

JEE Maths requires students to apply concepts from different fields of mathematics to find solutions effectively. Problems like vector subtraction in coordinate geometry, as in the given exercise, are typical examples wherein a JEE aspirant should proficiently translate geometric vectors into algebraic forms to compare with given conditions or options.

For successful preparation and approach to JEE vectors problems, it's crucial to practice the abstraction of mathematical concepts into applied problems, and to do so, understanding the coordinate geometry fundamentals and vector operations is essential. This solidifies the connection between abstract mathematical theory and practical problem-solving skills required for JEE and other advanced mathematics exams.