Problem 12 - Precalculus

Question

The coordinates of two points \(P\) and \(Q\) are given. In case, determine the components of the vector \(\overrightarrow{P Q}\). Write your answers in the form \(\langle a, b\rangle .\) $$P(1,0) \text { and } Q(0,1)$$

Step-by-Step Solution

Answer

The components of the vector \(\overrightarrow{PQ}\) are \(\langle -1, 1 \rangle\).

Step 1: Understanding the Points

First, identify the coordinates of the given points. The coordinates of point \(P\) are \((1,0)\), and the coordinates of point \(Q\) are \((0,1)\).

Step 2: Vector Definition

Recall that the vector \(\overrightarrow{PQ}\) represents the direction and distance from point \(P\) to point \(Q\). It is calculated by subtracting the coordinates of point \(P\) from the coordinates of point \(Q\), following the formula: \(\overrightarrow{PQ} = (x_2 - x_1, y_2 - y_1)\).

Step 3: Substituting Coordinates

Substitute the coordinates \((x_1, y_1) = (1, 0)\) for point \(P\), and \((x_2, y_2) = (0, 1)\) for point \(Q\) into the vector formula: \(\overrightarrow{PQ} = (0 - 1, 1 - 0)\).

Step 4: Calculating the Components

Perform the subtraction for both the x and y coordinates to find the components of the vector: \(\overrightarrow{PQ} = (-1, 1)\). So, the vector has components \(\langle -1, 1 \rangle\).

Step 5: Result

The components of the vector \(\overrightarrow{PQ}\) are \(\langle -1, 1 \rangle\).

Key Concepts

Coordinate Geometry

Coordinate geometry is the study of geometrical figures using a coordinate system, usually the Cartesian plane. In this exercise, we identify the points by their coordinates. A point in coordinate geometry is represented by a pair of numbers. The first number, often known as "x," shows the position on the horizontal axis, while the second number, "y," indicates the position on the vertical axis. For example:

• The point \(P(1, 0)\) means that it is located 1 unit along the x-axis and 0 units along the y-axis.
• The point \(Q(0, 1)\) is 0 units on the x-axis and 1 unit on the y-axis.

Using coordinates makes it easy to calculate distances between points and determine vector components. When given two points, plotting them on the plane can help visualize their relative positions, which is a foundational skill in coordinate geometry, useful for calculating vector components.

Vector Calculation

A vector in mathematics is an object that has both a direction and a magnitude. In this context, the vector \(\overrightarrow{PQ}\) points from point \(P\) to point \(Q\). Calculating the vector components involves simple arithmetic. You subtract the coordinates of the initial point from those of the terminal point. This is done both for the x-component and the y-component of the vector.

• For the x-component: \(x_2 - x_1 = 0 - 1 = -1\)
• For the y-component: \(y_2 - y_1 = 1 - 0 = 1\)

Hence, the vector \(\overrightarrow{PQ}\) is represented as \(\langle -1, 1 \rangle\). Vector components give us important information about the change in position from one point to another in coordinate systems, crucial for vector analysis.

Vectors in the Plane

Vectors are fundamental in expressing directions and magnitudes in a plane. A plane is a flat, two-dimensional surface where you can graph vectors using coordinate points. The components of a vector tell us how far and in which direction we have to move from the start point to reach the endpoint. In the given problem, the vector \(\overrightarrow{PQ}\) implies moving 1 unit left and 1 unit up.

• Moving \(-1\) along the x-axis signifies going to the left of the starting point.
• Moving \(+1\) along the y-axis represents movement upwards from the starting point.

Recognizing vectors in the plane helps in many fields such as physics, engineering, and computer science, where understanding motion and forces is necessary. Thus, grasping vector components and their functions in coordinate geometry enhances spatial comprehension and problem-solving skills.