Problem 7 - Precalculus

Question

The coordinates of two points \(P\) and \(Q\) are given. In case, determine the components of the vector \(\overrightarrow{P Q}\). Write your answers in the form \(\langle a, b\rangle .\) $$P(2,3) \text { and } Q(3,7)$$

Step-by-Step Solution

Answer

The vector \(\overrightarrow{PQ}\) is \(\langle 1, 4 \rangle\).

Step 1: Identify the Points

The problem provides two points, \(P\) and \(Q\), with the coordinates \(P(2, 3)\) and \(Q(3, 7)\). Our goal is to find the vector from \(P\) to \(Q\).

Step 2: Apply the Vector Formula

To find the vector \(\overrightarrow{PQ}\), use the formula: \(\overrightarrow{PQ} = (x_2 - x_1, y_2 - y_1)\). Here, \((x_1, y_1)\) are the coordinates of \(P\) and \((x_2, y_2)\) are the coordinates of \(Q\).

Step 3: Subtract the Coordinates

Apply the formula by calculating each component separately: \(x_2 - x_1 = 3 - 2\) and \(y_2 - y_1 = 7 - 3\).

Step 4: Calculate the Vector Components

The calculations give \(x_2 - x_1 = 1\) and \(y_2 - y_1 = 4\). Thus, the vector \(\overrightarrow{PQ}\) is \(\langle 1, 4 \rangle\).

Step 5: Express the Vector

The vector \(\overrightarrow{PQ}\) is expressed in component form as \(\langle 1, 4 \rangle\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that allows us to use algebraic equations to describe and analyze geometric shapes and figures. In the context of this exercise, we use the coordinate plane, consisting of the x-axis and y-axis, to determine the position of points. By providing coordinates such as \( P(2, 3) \) and \( Q(3, 7) \), we can pinpoint exact locations within this two-dimensional space.

The coordinates are typically ordered pairs \((x, y)\), where \(x\) represents the horizontal distance from the origin (0, 0) and \(y\) the vertical distance. Understanding coordinate geometry is crucial for translating complex mathematical concepts into visual representations. It helps simplify the process of finding the distance between points, understanding lines' slopes, and even performing operations like vector addition or subtraction.

Vectors

Vectors are mathematical entities with both magnitude and direction. Unlike scalars, which only have magnitude, vectors are often represented visually as arrows in the coordinate plane. The length of the arrow indicates the vector's magnitude, while the arrow's direction shows its orientation.

In this exercise, the vector \(\overrightarrow{PQ}\) is determined by the coordinates of points \(P\) and \(Q\). Vectors are crucial in physics and engineering because they help describe quantities like displacement, velocity, and force, which naturally have both size and direction. It's important to note that vectors can be summed, subtracted, or even multiplied, which makes them incredibly versatile tools in mathematical problem-solving.

Vector Subtraction

Vector subtraction is used to find the vector connecting two points and represents the direction and distance from one point to another. To subtract vectors, particularly in coordinate geometry, you subtract the respective components.
For instance, to find \(\overrightarrow{PQ}\), we use the formula: \(\overrightarrow{PQ} = (x_2 - x_1, y_2 - y_1)\). By substituting the given coordinates \(P(2, 3)\) and \(Q(3, 7)\) into the formula, we find the components of the vector to be \(\langle 1, 4 \rangle\).

• The difference \((x_2 - x_1)\) corresponds to movement along the x-axis.
• The difference \((y_2 - y_1)\) corresponds to movement along the y-axis.

This subtraction process essentially tells you how far and in what direction you need to travel from point \(P\) to reach point \(Q\), making it a fundamental operation in vector analysis.