Problem 16 - Algebra and Trigonometry

Question

\(9-18\) . Express the vector with initial point \(P\) and terminal point \(Q\) in component form. $$ P(-1,3), \quad Q(-6,-1) $$

Step-by-Step Solution

Answer

The vector in component form is \((-5, -4)\).

Step 1: Understand the Problem

We need to find the vector in component form given its initial point \( P(-1, 3) \) and terminal point \( Q(-6, -1) \).

Step 2: Recall the Formula

The component form of a vector is given by \((Q_x - P_x, Q_y - P_y)\), where \(P(x,y)\) is the initial point and \(Q(x,y)\) is the terminal point.

Step 3: Substitute the Coordinates

Substitute the coordinates of the points \(P(-1, 3)\) and \(Q(-6, -1)\) into the formula: \((-6 - (-1), -1 - 3)\).

Step 4: Calculate the Differences

Calculate each component: 1. For the x-component: \(-6 - (-1) = -6 + 1 = -5\)2. For the y-component: \(-1 - 3 = -4\).

Step 5: Write the Component Form

Combine the calculated components into the vector's component form: \((-5, -4)\).

Key Concepts

Understanding the Initial Point

An **initial point** in vector mathematics refers to the starting location of a vector. This point is usually denoted as \( P(x, y) \). It is the reference from which the vector begins.In our given problem, the initial point is \( P(-1, 3) \).

• The x-coordinate of the initial point is \(-1\).
• The y-coordinate of the initial point is \(3\).

Remember, the coordinates of the initial point help to determine the vector's direction and length. You always start from the initial point when finding the component form of a vector.

Defining the Terminal Point

A **terminal point** is where the vector ends. It is sometimes referred to as the endpoint.This is usually denoted as \( Q(x, y) \). In our example, the terminal point is \( Q(-6, -1) \).

• The x-coordinate of the terminal point is \(-6\).
• The y-coordinate of the terminal point is \(-1\).

The terminal point serves as the destination in vector operations. The difference between the initial and terminal points determines the vector's component form.

Using the Vector Formula

The **vector formula** is key to converting a vector from its initial and terminal points to component form. This formula is written as: \[ \text{Vector } \overrightarrow{PQ} = (Q_x - P_x, Q_y - P_y) \]Here:

• \( Q_x \) and \( Q_y \) are the x and y coordinates of the terminal point.
• \( P_x \) and \( P_y \) are the x and y coordinates of the initial point.

By using this formula, we can compute how much we "move" from the initial to the terminal point by subtracting their corresponding coordinates.

The Role of Coordinate Subtraction

**Coordinate subtraction** is the process by which we find the difference between the coordinates of the terminal point and the initial point. This difference gives us the vector's component form.To understand how this works, let's take the coordinates from our problem:

• For the x-component: \(-6 - (-1) = -6 + 1 = -5\)
• For the y-component: \(-1 - 3 = -4\)

The subtraction results in the vector's component form \((-5, -4)\).This form shows how much we have to "move" along the x-axis and y-axis from the initial point to reach the terminal point. It's crucial in defining the vector's direction and magnitude in a coordinate plane.