Problem 8 - 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 \(\langle a, b\rangle .\) $$P(5,1) \text { and } Q(4,9)$$

Step-by-Step Solution

Answer

The vector \(\overrightarrow{PQ}\) is \(\langle -1, 8 \rangle\).

Step 1: Identify the Coordinates of Points P and Q

The coordinates of point \(P\) are \((5, 1)\) and for point \(Q\) are \((4, 9)\).

Step 2: Use the Vector Formula

To find the vector \(\overrightarrow{PQ}\) from point \(P\) to point \(Q\), use the formula: \(\overrightarrow{PQ} = (x_2 - x_1, y_2 - y_1)\).

Step 3: Compute the Change in x-Coordinates

Calculate the change in the x-coordinates: \( x_2 - x_1 = 4 - 5 = -1 \).

Step 4: Compute the Change in y-Coordinates

Calculate the change in the y-coordinates: \( y_2 - y_1 = 9 - 1 = 8 \).

Step 5: Form the Vector Components

Combine the results from Steps 3 and 4 to obtain the vector \(\overrightarrow{PQ}\): \( \langle -1, 8 \rangle \).

Key Concepts

Coordinates

Coordinates are a fundamental part of geometry and vector calculations. They are used to specify the exact location of a point in a plane or space. In a two-dimensional plane, each point is represented by a pair of numbers, typically written as \((x, y)\). These numbers are known as the point's coordinates.
For example, the point \(P\) given as \((5, 1)\) tells us that the point is located 5 units along the x-axis and 1 unit along the y-axis.
Similarly, point \(Q\) is positioned at \((4, 9)\).
Coordinates help us understand and visualize where a point is placed on a graph, and they are essential in determining relationships between points, such as finding the direction and distance from one point to another.

Vector Calculation

Vector calculation involves determining quantities that have both direction and magnitude. When finding the vector between two points, we particularly focus on how to navigate from one to the other in terms of horizontal and vertical changes.
To find the vector from point \(P\) to point \(Q\), we use the formula \(\overrightarrow{PQ} = (x_2 - x_1, y_2 - y_1)\).
This tells us that the vector components are calculated by subtracting the coordinates of the starting point from the coordinates of the endpoint.
Vectors provide a way to quantify which way you need to move from point \(P\) to point \(Q\) and by how much.

Point Coordinates

Point coordinates provide the exact location of a point in relation to an origin. These coordinates are crucial when working with vectors, as they allow us to determine the starting and ending points of a vector.
For any point \((x, y)\), \(x\) denotes the position along the x-axis (horizontal), and \(y\) denotes the position along the y-axis (vertical).
In our exercise, we started with points \(P(5, 1)\) and \(Q(4, 9)\).
Understanding how to read and interpret point coordinates is key to finding changes and calculating vectors, as they are the initial data points required for vector computation.

Change in Coordinates

The change in coordinates involves calculating the difference in position along each axis when moving from one point to another. This change is what forms the components of a vector.
To determine the horizontal component (change in x):

• Subtract the x-coordinate of point \(P\) from the x-coordinate of point \(Q\).
• In our example: \(x_2 - x_1 = 4 - 5 = -1\).

The vertical component (change in y) involves:

• Subtracting the y-coordinate of point \(P\) from point \(Q\).
• For this problem: \(y_2 - y_1 = 9 - 1 = 8\).

These changes give us the vector \(\overrightarrow{PQ} = \langle -1, 8 \rangle\), showing how much you move horizontally and vertically to go from \(P\) to \(Q\).
Understanding the change in coordinates is crucial for grasping how vectors represent both direction and magnitude in two-dimensional space.