State the position vectors of the points with coordinates (a) \(\mathrm{P}(4,7)\), (b) \(\mathrm{Q}(-3,5)\), (c) \(\mathrm{R}(0,3)\), (d) \(\mathrm{S}(-1,0)\).

In the realm of vector mathematics, we explore quantities that have both magnitude and direction. These are distinct from scalar quantities, which have only magnitude. A key component of vector mathematics is the 'position vector', which represents the position of a point in space relative to an origin point.

To describe a position vector in a two-dimensional plane, we use an ordered pair of numbers, which correspond to coordinates. These coordinates are like instructions that tell us how far to move along the x-axis (horizontal movement) and the y-axis (vertical movement) from the origin to reach our desired point. A position vector will often be denoted in unit vector notation, where \(\hat{i}\) represents a unit vector in the direction of the x-axis, and \(\hat{j}\) represents a unit vector in the direction of the y-axis.

For instance, the point P(4,7) in the given exercise defines a position vector as \(\vec{P} = 4\hat{i} + 7\hat{j}\). In this notation, the coefficients of \(\hat{i}\) and \(\hat{j}\) (4 and 7, respectively) denote the vector's magnitude in the directions of the respective axes. These simple notations are elegantly powerful, enabling complex operations like addition, subtraction, and scaling of vectors, which are imperative in many fields of science and engineering.

Coordinate geometry is the study of geometric figures graphed on a coordinate plane. This branch of mathematics is essential in bridging algebra with geometry, allowing complex problems to be solved visually and algebraically.

Points on a plane are denoted by pairs of numeric coordinates, which are essentially distances measured from two perpendicular axes intersecting at an origin, typically labeled as (0,0). For example, the point Q(-3,5) can be located by moving 3 units to the left and 5 units up from the origin.

In the exercise provided, each point specified has a unique position described by its coordinates. These coordinates inform the construction of the position vectors we've talked about, translating geometric points into vectorial algebra. This translation facilitates calculations in various applications, ranging from computer graphics to physics, where precise positioning is crucial.

Mathematics is an indispensable tool for engineers, providing the language and methods needed to model and solve problems in the physical world. One critical aspect of this is understanding how to work with vectors, as they commonly represent forces, velocities, displacements, and other engineering quantities in both two and three dimensions.

Engineers utilize vectors to describe mechanical systems accurately, forecast behaviors, design structures, and much more. Position vectors, like those determined in the exercise, are foundational in representing points in space, which can be related to the position of a component in a mechanical design or the coordinate of a structural element in architectural engineering.

The notations and principles from vector mathematics and coordinate geometry merge here, allowing engineers to simulate, analyze, and optimize real-world systems. Clear comprehension of these concepts aids engineers in pushing the boundaries of innovation while ensuring safety and effectiveness in their design solutions.