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A position vector is a vector that denotes a specific location in space. It is often depicted in terms of components, such as along the horizontal and vertical axes in a coordinate system. For example, the position vector of a point in a two-dimensional plane is represented as \( a \mathbf{i} + b \mathbf{j} \), where \( a \) and \( b \) are the coordinates of the point on the x-axis and y-axis, respectively.

Position vectors are fundamental in understanding the location of a point and providing a basis to perform vector operations, such as addition or subtraction. Since each position vector originates from the origin and extends to a point in the plane, it provides insight into how the point is positioned relative to the origin. In this exercise, the position vectors of points \(P(-4,1)\) and \(Q(3,-4)\) are \(\mathbf{P} = -4 \mathbf{i} + 1 \mathbf{j}\) and \(\mathbf{Q} = 3 \mathbf{i} - 4 \mathbf{j}\) respectively. This representation frames the points in a way that sets the stage for operations, such as vector subtraction, that can determine directional relationships between points.

Vector subtraction is an essential operation when dealing with vectors. It allows you to find the direction and magnitude from one point to another in space. To subtract one vector from another, you simply subtract the corresponding components of the vectors.

In the context of position vectors, subtracting the vector of one point from the vector of another gives you the vector that points from the first to the second point. For instance, to find vector \( \mathbf{PQ} \), which represents the vector from point \( P \) to point \( Q \), we perform the subtraction: \( \mathbf{PQ} = \mathbf{Q} - \mathbf{P} \).

Here's a breakdown of our exercise:

• The position vector of \(Q\) is \(3 \mathbf{i} - 4 \mathbf{j}\).
• The position vector of \(P\) is \(-4 \mathbf{i} + 1 \mathbf{j}\).
• We calculate \( \mathbf{PQ} = (3 \mathbf{i} - 4 \mathbf{j}) - (-4 \mathbf{i} + 1 \mathbf{j}) \).

Performing this subtraction, we get \( 7 \mathbf{i} - 5 \mathbf{j} \), which gives us both the direction and the "distance" from \(P\) to \(Q\).

Coordinate geometry, also known as analytic geometry, is a field of mathematics that uses algebraic equations to describe and analyze geometric shapes and their properties in a coordinate system.

This system allows the use of coordinates to uniquely position points in the plane, which is integral for solving many geometric problems. Through the coordinate system, the connections and interactions of geometric figures, like lines and curves, with algebraic equations are established.

For example, consider points \(P(-4,1)\) and \(Q(3,-4)\) in the exercise. These points are identified with a coordinate pair indicating their location on the plane relative to the axes. The application of vector operations, such as finding the vector \(\mathbf{PQ}\), becomes feasible through the simplicity of these coordinate representations.

Using coordinate geometry, we also define geometric features like distances, midpoints, and slopes through coordinate systems, enhancing our ability to comprehend and demonstrate geometric relationships via vector operations in a straightforward manner.