Problem 10 - Thomas Calculus

Question

In Exercises \(9-16,\) find the component form of the vector. The vector \(\overrightarrow{O P}\) where \(O\) is the origin and \(P\) is the midpoint of segment \(R S,\) where \(R=(2,-1)\) and \(S=(-4,3)\)

Step-by-Step Solution

Answer

Vector \(\overrightarrow{O P}\) is \((-1, 1)\).

Step 1: Understand the Problem

We need to find the vector \(\overrightarrow{O P}\), where \(O\) is the origin \((0,0)\) and \(P\) is the midpoint of the segment \(R S\). Points \(R\) and \(S\) are given as \((2,-1)\) and \((-4,3)\), respectively.

Step 2: Find Midpoint of Segment RS

To find the midpoint \(P\) of segment \(R S\), use the midpoint formula \((x,y) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\). Substitute \((x_1, y_1) = (2, -1)\) and \((x_2, y_2) = (-4, 3)\): \[\left(\frac{2 + (-4)}{2}, \frac{-1 + 3}{2}\right) = \left(\frac{-2}{2}, \frac{2}{2}\right) = (-1, 1)\].Thus, the midpoint \(P\) is \((-1, 1)\).

Step 3: Find Component Form of Vector OP

The component form of vector \(\overrightarrow{O P}\) is found by subtracting the coordinates of \(O\) from the coordinates of \(P\). Since \(O = (0,0)\) and \(P = (-1,1)\), the vector \(\overrightarrow{O P}\) is:\[(-1 - 0, 1 - 0) = (-1, 1)\].

Step 4: Express the Answer

Hence, the component form of the vector \(\overrightarrow{O P}\) is \((-1, 1)\).

Key Concepts

Understanding the Midpoint Formula

The midpoint formula is crucial for finding the center point of a line segment defined by two end points. This formula helps in determining the exact point that lies halfway between two given points on a coordinate plane. It uses the average of the x-coordinates and the average of the y-coordinates from the endpoints to find this midpoint.
For example, if you have two points \( R(x_1, y_1) \) and \( S(x_2, y_2) \), the formula for the midpoint \( P \) is given by:\[(x, y) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\]Applying this formula is straightforward:

• Add the x-coordinates of both points together and divide by 2 to find the x-coordinate of the midpoint.
• Add the y-coordinates of both points together and divide by 2 to find the y-coordinate of the midpoint.

This formula provides a simple yet effective method for calculating the midpoint. It's a fundamental tool in geometry and vector calculus for tasks like dividing segments, analyzing shapes, and finding symmetry.

Component Form of a Vector

The component form of a vector simplifies how we work with vectors by expressing them in terms of their horizontal and vertical components. This form is essential for distinguishing vectors from simple coordinate points as it highlights the direction and magnitude from one point to another.
To determine the component form of a vector \( \overrightarrow{AB} \) given two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \), you calculate:\[(x_2 - x_1, y_2 - y_1)\] This translates to:

• Subtract the x-coordinate of the initial point from the x-coordinate of the terminal point to get the x-component.
• Subtract the y-coordinate of the initial point from the y-coordinate of the terminal point to get the y-component.

In practical terms, the component form makes it clear how much a vector "moves" horizontally and vertically, breaking it down into manageable parts for further vector operations.

Basics of Vector Operations

Vector operations are significant in understanding how to manipulate and utilize vectors in both two-dimensional and three-dimensional spaces. These operations include addition, subtraction, and scalar multiplication, providing the foundation for more complex calculations.

• Addition: To add two vectors, simply add their respective components. If \( \overrightarrow{v} = (v_1, v_2) \) and \( \overrightarrow{u} = (u_1, u_2) \), then the sum is \( \overrightarrow{v} + \overrightarrow{u} = (v_1 + u_1, v_2 + u_2) \).
• Subtraction: Subtract the corresponding components of one vector from another. For the same vectors mentioned above, the difference is \( \overrightarrow{v} - \overrightarrow{u} = (v_1 - u_1, v_2 - u_2) \).
• Scalar Multiplication: Multiply each component of the vector by the scalar value. If \( c \) is a scalar and \( \overrightarrow{v} = (v_1, v_2) \), then \( c \cdot \overrightarrow{v} = (c \cdot v_1, c \cdot v_2) \).

These basic operations are vital in problem-solving scenarios such as calculating forces, optimizing paths, and many areas of physics and engineering. Understanding and performing these vector operations accurately allows you to analyze and model real-world phenomena efficiently.