Problem 10 - University Calculus: Early Transcendentals

Question

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

The component form of the vector is (-1, 1).

Step 1: Identify the Coordinates of R and S

The problem states the coordinates of points \(R\) and \(S\) are \(R = (2, -1)\) and \(S = (-4, 3)\) respectively. These will be used to find the midpoint.

Step 2: Calculate the Midpoint of Segment RS

The midpoint \(P\) of a segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula: \(P = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\). For \(R = (2, -1)\) and \(S = (-4, 3)\), the midpoint \(P\) is: \(P = \left( \frac{2 + (-4)}{2}, \frac{-1 + 3}{2} \right) = (-1, 1)\).

Step 3: Determine the Component Form of Vector OP

The vector \(\overrightarrow{OP}\) starting at the origin \(O = (0, 0)\) and ending at \(P = (-1, 1)\) is expressed in component form by subtracting the coordinates of the initial point from the coordinates of the terminal point. Therefore, \(\overrightarrow{OP} = (x_2 - x_1, y_2 - y_1) = (-1 - 0, 1 - 0) = (-1, 1)\).

Key Concepts

Understanding Component Form

In vector mathematics, the component form of a vector is crucial for understanding its direction and magnitude. Imagine a vector as an arrow pointing from one point to another in a coordinate system. The component form is essentially a pair of numbers that describe how far that arrow reaches in the horizontal ( x-axis) and vertical ( y-axis) directions. To express a vector in component form, you simply subtract the coordinates of the starting point from the coordinates of the endpoint:

• The difference in x-coordinates gives you the horizontal component.
• The difference in y-coordinates gives you the vertical component.

So, if you have a vector \(\overrightarrow{AB}\) with starting point \(A(x_1, y_1)\) and endpoint \(B(x_2, y_2)\), the component form is given by:\[\overrightarrow{AB} = (x_2 - x_1, y_2 - y_1)\]This formula allows you to see how the vector "moves" between the points, revealing both direction and scale with respect to the components.

Essentials of Midpoint Calculation

Calculating the midpoint of a segment in coordinate geometry is all about finding the exact middle point between two endpoints. It’s like determining the center spot between two locations on a map. Midpoint calculation is not only straightforward but also useful for various applications such as locating the center of a line segment.The formula for calculating the midpoint \(P\) of a segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is:\[P = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]This involves simply averaging the corresponding coordinates of the endpoints:

• Add the x-coordinates of both points, then divide by 2.
• Add the y-coordinates of both points, then divide by 2.

This process results in a new set of coordinates that represent the midpoint, balancing the segment evenly between the original points.

The Basics of Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is all about representing geometric shapes and their relationships using algebraic equations. It’s like using a map, where every point is defined by a set of coordinates.In a typical coordinate plane:

• The x-axis runs horizontally.
• The y-axis runs vertically.
• Every point is represented as a pair \((x, y)\).

This framework allows for the graphical representation of equations and the ability to find distances, midpoints, slopes, and angles between lines. Understanding coordinate geometry helps you visualize mathematical problems and solve them much more intuitively.For example, if you know the coordinates of two points, you can use them to form a line—each aspect of this line, from slope to intercepts, can be analyzed through simple calculations.