Problem 14 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The component form of the vector is \((8, 8)\).

Step 1: Understanding the Problem

To express the vector in component form, we need to identify the initial and terminal points of the vector. The initial point is given as \( P(1,1) \) and the terminal point is \( Q(9,9) \).

Step 2: Compute the Components

The component form of a vector with initial point \( P(x_1, y_1) \) and terminal point \( Q(x_2, y_2) \) is given by the formula: \( (x_2 - x_1, y_2 - y_1) \). Here, \( x_1 = 1 \), \( y_1 = 1 \), \( x_2 = 9 \), and \( y_2 = 9 \).

Step 3: Calculate the Differences

Subtract the coordinates of \( P \) from the coordinates of \( Q \) to find the components. Calculate \( x_2 - x_1 = 9 - 1 = 8 \) and \( y_2 - y_1 = 9 - 1 = 8 \).

Step 4: Write the Vector in Component Form

Using the calculated differences from the previous step, express the vector in component form as \( (8, 8) \).

Key Concepts

Initial and Terminal Points in Vector Calculations

When calculating a vector in component form, it's important to first understand the roles of the initial and terminal points. In mathematics, every vector is defined by these two points. The initial point, often denoted as \(P(x_1, y_1)\), is where the vector begins. The terminal point, \(Q(x_2, y_2)\), is where the vector ends.
Understanding these points is crucial since they determine the direction and length of the vector. In our example, the initial point is \(P(1,1)\) and the terminal point is \(Q(9,9)\). These coordinates are like the start and end locations in a journey. The vector shows the path taken from start to finish. Recognizing these two points helps you set the stage for the vector's calculation in component form.

Understanding Coordinate Subtraction

Once you've identified the initial and terminal points, the next step involves calculating the vector components. This is done through coordinate subtraction. It's a simple yet crucial process. The component form of a vector is essentially the "difference" in the \(x\) and \(y\) coordinates between the terminal and initial points.
Here's how you do it:

• Subtract the \(x\)-coordinate of the initial point from the \(x\)-coordinate of the terminal point: \(x_2 - x_1\).
• Subtract the \(y\)-coordinate of the initial point from the \(y\)-coordinate of the terminal point: \(y_2 - y_1\).

In our given problem, the coordinate subtraction is \(9 - 1\) for both the \(x\) and \(y\) coordinates. This results in the components \(8\) and \(8\). Each subtraction tells you how far, and in which direction, you've traveled just in those dimensions.

Vector Calculation Steps Demystified

Calculating the vector component form is straightforward if you follow a few clear steps. Here's a simplified breakdown:

• Identify the initial and terminal points: These are \(P\) and \(Q\) in your problem.
• Perform coordinate subtraction to find the difference in coordinates: Use \(x_2 - x_1\) and \(y_2 - y_1\).
• Write the vector in component form using the differences obtained: If your results are \(8\) in the x-direction and \(8\) in the y-direction, your vector is \((8, 8)\).

Through these steps, you translate geometric details into a numerical form that defines the vector's direction and magnitude. Remember, the simplicity of these steps does not diminish their power in defining the vector precisely.