Problem 13 - Precalculus Mathematics for Calculus

Question

Express the vector with initial point \(P\) and terminal point \(Q\) in component form. $$P(3,2), \quad Q(8,9)$$

Step-by-Step Solution

Answer

The vector in component form is \( \langle 5, 7 \rangle \).

Step 1: Understand the Problem

We need to express the vector that starts at point \( P(3,2) \) and ends at point \( Q(8,9) \) in component form. A vector in component form is written as \( \langle x_2 - x_1, y_2 - y_1 \rangle \), where \( (x_1, y_1) \) are the coordinates of the initial point and \( (x_2, y_2) \) are the coordinates of the terminal point.

Step 2: Identify Initial and Terminal Points

The coordinates of the initial point \( P \) are \( (3, 2) \), and the coordinates of the terminal point \( Q \) are \( (8, 9) \).

Step 3: Calculate the Differences

Find the difference between the corresponding coordinates of the terminal point \( Q \) and the initial point \( P \): \[ x_2 - x_1 = 8 - 3 = 5 \]\[ y_2 - y_1 = 9 - 2 = 7 \]

Step 4: Write the Component Form of the Vector

The component form of the vector is the pair \( \langle x_2 - x_1, y_2 - y_1 \rangle \). Using our calculated differences, the vector is \( \langle 5, 7 \rangle \).

Key Concepts

Initial Point

In the context of vectors, an **initial point** is where a vector begins. It's the starting coordinate that helps us understand where a vector originates. For instance, if a vector describes a journey, the initial point is like your starting point on a map.

When given in a coordinate plane, as in this exercise, the initial point is described by its

• x-coordinate
• y-coordinate

These create an ordered pair, usually noted as \( P(x_1, y_1) \).

In our exercise, the initial point is \( P(3, 2) \). This means the vector starts at the coordinate where x is 3 and y is 2.

Terminal Point

A **terminal point** is where a vector finishes, essentially acting as the endpoint. It represents the final position in the journey depicted by the vector.

Much like the initial point, it is denoted by an ordered pair, \( Q(x_2, y_2) \), which marks its position on the coordinate plane.

For our exercise, the terminal point is \( Q(8, 9) \). This tells us that the vector ends where x equals 8 and y equals 9.

Coordinate Differences

Understanding the **coordinate differences** between the terminal and initial points is crucial for finding a vector's component form. The differences tell us how far the vector has moved from its start to end point in each direction.

For any vector, these differences are calculated as:

• The difference in x-coordinates: \( x_2 - x_1 \)
• The difference in y-coordinates: \( y_2 - y_1 \)

This subtraction helps determine the vector's direction and magnitude along the x and y axes.

In our scenario:

• The x-coordinate difference is \( 8 - 3 = 5 \)
• The y-coordinate difference is \( 9 - 2 = 7 \)

These values show the vector moves 5 units in the x-direction and 7 units in the y-direction.

Component Vector Calculation

Once the coordinate differences are found, we can express the vector in **component form**. This form clearly illustrates the movement from the initial to the terminal point and is typically written as \( \langle x_2 - x_1, y_2 - y_1 \rangle \).

In component form, each value inside the brackets signifies the vector's movement

• The first value for the x-axis
• The second value for the y-axis

For our exercise, the component vector is \( \langle 5, 7 \rangle \), which means the vector moves 5 units along the x-axis and 7 units along the y-axis.

Understanding component form is essential in mathematics and physics, as it breaks down a vector into its primary directions, making it easier to analyze and apply in various problems.