Problem 9 - 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 \(\langle a, b\rangle .\) $$P(-2,-3) \text { and } Q(-3,-2)$$

Step-by-Step Solution

Answer

The vector is \(\langle -1, 1 \rangle\).

Step 1: Understanding the Problem

We need to find the components (or direction) of the vector \(\overrightarrow{PQ}\). The components of a vector are given by the difference in the x-coordinates and the y-coordinates of the two points \(P\) and \(Q\).

Step 2: Finding the Difference in x-coordinates

Subtract the x-coordinate of point \(P\) from the x-coordinate of point \(Q\). So, we have: \(-3 - (-2) = -3 + 2 = -1\).

Step 3: Finding the Difference in y-coordinates

Subtract the y-coordinate of point \(P\) from the y-coordinate of point \(Q\). So, we have: \(-2 - (-3) = -2 + 3 = 1\).

Step 4: Writing the Vector Components

Now, combine the differences in the x-coordinates and y-coordinates to get the vector \(\overrightarrow{PQ}\) in component form. The vector is \(\langle -1, 1 \rangle\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This field allows for precise definitions and manipulations of geometric shapes using algebraic symbols. In our exercise, we deal with two points, \(P(-2, -3)\) and \(Q(-3, -2)\), which are located on a Cartesian plane. The coordinates \((x, y)\) represent distances along the horizontal and vertical axes, respectively. In coordinate geometry, points, lines, and other figures are defined using coordinates, which allows for effective analysis and transformation in space. This approach provides tools for solving complex geometric problems through algebraic methods. Understanding the coordinates of points is essential for calculating distances, midpoints, and other properties like vector components.

Vector Subtraction

Vector subtraction is a key concept when working with vectors. When you subtract vectors, you essentially find the change from one point to another in a space, described by the coordinates given.In our example with points \(P\) and \(Q\), vector subtraction involves calculating the components of the vector \(\overrightarrow{PQ}\). We subtract the coordinates of \(P\) from \(Q\) for both x and y dimensions:

• For the x-component: \(-3 - (-2) = -1\)
• For the y-component: \(-2 - (-3) = 1\)

These steps provide the "delta," or change, from point \(P\) to point \(Q\). The resulting vector \(\langle -1, 1 \rangle\) tells us the movement required in the x and y directions to go from \(P\) to \(Q\). This is a fundamental operation in physics and engineering, reflecting changes in position or other quantities.

Direction of Vectors

The direction of a vector is described by its components, which indicates how much the vector moves along each axis. In simpler terms, it tells you where the vector is pointing in a plane.For our vector \(\overrightarrow{PQ}\) with components \(\langle -1, 1 \rangle\), we can interpret the direction it gives. The negative x-component \(-1\) suggests a movement leftward on the horizontal axis, while the positive y-component \(1\) indicates a movement upward on the vertical axis. Together, these directional shifts create a path from point \(P\) to point \(Q\).Understanding direction is crucial when we want to comprehend not just magnitude but also the orientation of changes or forces in systems. Proper appreciation of vector directions aids in mapping paths and forces in various applications across science and engineering.