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Vector components are the building blocks of any vector in a multi-dimensional space. When dealing with points on a plane, as in coordinate geometry, each vector can be broken down into its individual components along the x and y axes. For instance, when determining the vector from point \(P\) to point \(Q\), you consider the difference in the x-coordinates and the y-coordinates individually.

• Subtract the x-coordinate of the initial point from the x-coordinate of the terminal point to find the x-component.
• Subtract the y-coordinate of the initial point from the y-coordinate of the terminal point to find the y-component.

For \(\overrightarrow{PQ}\) from the given exercise:- The x-component is calculated as \(1 - (-1) = 2\).- The y-component is calculated as \(5 - 3 = 2\).Putting these together, the vector \(\overrightarrow{PQ}\) in component form is \((2, 2)\). This method helps break down the vector into manageable parts that are easy to analyze and use in further calculations.

Unit vectors are vectors with a magnitude of one, used to define directions in a coordinate system without adding scale. In two-dimensional space, the standard unit vectors are commonly denoted as \( \mathbf{i} \) and \( \mathbf{j} \).- \( \mathbf{i} \) represents a unit vector along the x-axis.- \( \mathbf{j} \) represents a unit vector along the y-axis.In the exercise provided, after determining the components of the vector \(\overrightarrow{PQ} - \overrightarrow{PR}\) to be \((4, -2)\), we can express it using these unit vectors:Convert each component into a multiplication with the corresponding unit vector:- The x-component "4" becomes \(4\mathbf{i}\).- The y-component "-2" becomes \(-2\mathbf{j}\).Therefore, the vector \(\overrightarrow{PQ} - \overrightarrow{PR}\) can be written as \(4\mathbf{i} - 2\mathbf{j}\). This expression highlights the direction and relative magnitude of the vector in an easily visualizable format.

Coordinate geometry is a branch of mathematics that allows us to represent geometric figures and their relationships in a numerical form. By using a coordinate system, we can describe points as ordered pairs, like \((x, y)\), and vectors as connections between these points.When working with points like \(P(-1,3)\), \(Q(1,5)\), and \(R(-3,7)\), you use their coordinates to perform operations such as addition, subtraction, and finding vector directions and magnitudes. In our original exercise, we subtract vectors \(\overrightarrow{PR}\) from \(\overrightarrow{PQ}\) using the principles of coordinate geometry:- Calculating the differences between the respective components directly reflects the vector subtraction.This helps us not only to determine new vectors but also to explore spatial relationships. Understanding coordinate geometry enables you to solve problems related to distance, angle, and alignment between points and lines in a mathematical plane, providing a robust toolset for both theoretical and practical applications.