Problem 11 - 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 \(\langle a, b\rangle .\) $$P(-5,1) \text { and } Q(3,-4)$$

Step-by-Step Solution

Answer

The vector \( \overrightarrow{P Q} \) is \( \langle 8, -5 \rangle \).

Step 1: Identify Coordinates

First, identify the given coordinates for points \( P \) and \( Q \). We have \( P(-5, 1) \) and \( Q(3, -4) \).

Step 2: Apply Vector Formula

To find the vector \( \overrightarrow{P Q} \), subtract the coordinates of \( P \) from the coordinates of \( Q \) using the formula: \( \overrightarrow{P Q} = (x_2 - x_1, y_2 - y_1) \) where \( (x_1, y_1) \) are coordinates of \( P \) and \( (x_2, y_2) \) are coordinates of \( Q \).

Step 3: Subtract X-Coordinates

Calculate the difference of the x-coordinates: \( x_2 - x_1 = 3 - (-5) = 3 + 5 = 8 \).

Step 4: Subtract Y-Coordinates

Calculate the difference of the y-coordinates: \( y_2 - y_1 = -4 - 1 = -5 \).

Step 5: State Vector Components

The components of the vector \( \overrightarrow{P Q} \) is \( \langle 8, -5 \rangle \).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, combines algebra and geometry to describe and analyze the properties of figures using coordinates. In context, this involves plotting points on a plane defined by two perpendicular number lines, known as axes.
Each point on the plane is defined by a pair of numbers, called coordinates, which tell you where the point is located in relation to the origin, the point where the axes intersect.

• The first number in the pair, known as the \(x\)-coordinate, tells you how far to move horizontally from the origin.
• The second number, the \(y\)-coordinate, tells you how far to move vertically.

These pairs of numbers are written as \((x, y)\). In our exercise, points \(P\) and \(Q\) have coordinates \((-5, 1)\) and \((3, -4)\) respectively. Identifying these coordinates accurately is essential to solve problems involving vectors, as the location of each point determines the direction and length of the vector in coordinate geometry.

Vector Components

Vectors are mathematical entities that have both a magnitude (length) and a direction. They can be represented in component form, which consists of horizontal and vertical movements, akin to \(x\) and \(y\) shifts on a coordinate plane.
The component form of a vector from point \(P\) to \(Q\) is \ \(\overrightarrow{P Q} = \langle a, b \rangle\ \), where \(a\) and \(b\) are the vector's components. These components effectively break down the vector into simpler, understandable parts:

• \(a\): represents the change along the \(x\)-axis,
• \(b\): represents the change along the \(y\)-axis.

The components are calculated by subtracting the coordinates of the initial point \(P\) from the coordinates of the terminal point \(Q\). In this task, the vector \(\overrightarrow{P Q}\) was simplified to \(\langle 8, -5 \rangle\), clearly showing the precise changes from \(P\) to \(Q\) along each axis.

Subtraction

Subtraction in the context of vectors refers to finding the difference between corresponding coordinates. This method is utilized to determine the direction and length of a vector.
In mathematical terms, if you have two points \((x_1, y_1)\) and \((x_2, y_2)\), the vector \(\overrightarrow{P Q}\) is determined by subtracting the coordinates of point \(P\) from point \(Q\).

• First, subtract the \(x\)-coordinates: \(x_2 - x_1\).
• Then, subtract the \(y\)-coordinates: \(y_2 - y_1\).

These subtractions yield the vector components that we have in the component form, \(\langle 8, -5 \rangle\), with each component representing the distance and direction traveled from \(P\) to \(Q\). It is a straightforward subtraction operation, which shows how something changes as you move from one point to another on the plane.