Problem 30 - Algebra and Trigonometry

Question

The vector \(\mathbf{v}\) has initial point \(P\) and terminal point \(Q .\) Find its position vector. That is, express \(\mathbf{v}\) in the form \(a \mathbf{i}+b \mathbf{j} .\) $$ P=(-3,2) ; \quad Q=(6,5) $$

Step-by-Step Solution

Answer

\(\mathbf{v} = 9 \mathbf{i} + 3 \mathbf{j}\)

Step 1: Identify the components of the points

Extract the coordinates of the initial point \(P\) and the terminal point \(Q\). For point \(P\), the coordinates are \((-3,2)\). For point \(Q\), the coordinates are \((6,5)\).

Step 2: Determine the change in each coordinate

Calculate the differences between the corresponding coordinates of point \(Q\) and point \(P\). The formula is: \[\text{Change in } x = x_2 - x_1\] \[\text{Change in } y = y_2 - y_1\].

Step 3: Substitute the coordinates into the formulas

Using the coordinates of \(P(-3, 2)\) and \(Q(6, 5)\), we get: \[\text{Change in } x = 6 - (-3) = 6 + 3 = 9\].\[\text{Change in } y = 5 - 2 = 3\].

Step 4: Express the position vector

With the changes calculated, write the position vector \(\mathbf{v}\) in the form \(a \mathbf{i} + b \mathbf{j}\). The vector is: \[\mathbf{v} = 9 \mathbf{i} + 3 \mathbf{j}\].

Key Concepts

coordinate geometry

In coordinate geometry, we use a coordinate system to define the positions of points, lines, and shapes in a plane.
Each point in a 2-dimensional space can be identified using a pair of numbers, typically represented as \((x, y)\).
These numbers are known as the coordinates of the point.
The first number (\(x\)) is the x-coordinate and indicates the position along the horizontal axis, while the second number (\(y\)) is the y-coordinate and indicates the position along the vertical axis.
In this exercise, the points are given as \((-3, 2)\) for the initial point \(\text{P}\) and \((6, 5)\) for the terminal point \(\text{Q}\).
Using coordinate geometry, we can accurately measure and calculate the distance and direction between these two points by finding the difference in their coordinates, which is a key process in deriving the position vector.

vector components

Vectors are entities that have both magnitude and direction.
They are often represented by an arrow in a space, where the direction of the arrow indicates the direction of the vector and the length of the arrow represents its magnitude.
To work with vectors mathematically, we break them down into their horizontal and vertical parts, known as components.
Each vector in 2-dimensional space can be described in terms of its components along the x and y axes.
For our vector \(\textbf{v}\), the components are derived by calculating the differences in the x-coordinates and y-coordinates between the initial point \(\text{P}\) and the terminal point \(\text{Q}\).
Thus, we get the x-component by \(\text{Change in } x = x_2 - x_1 = 6 - (-3) = 9\) and the y-component by \(\text{Change in } y = y_2 - y_1 = 5 - 2 = 3\).
These components form the basis for expressing the vector in a standardized form.

vector notation

Vectors can be expressed in various notations, one of the common forms being the ij notation.
In this case, each vector is written as a linear combination of the unit vectors \(\textbf{i}\) and \(\textbf{j}\).
The unit vector \(\textbf{i}\) represents one unit along the x-axis, while \(\textbf{j}\) represents one unit along the y-axis.
So, any vector \(\textbf{v}\) in 2D can be expressed as \(a \textbf{i} + b \textbf{j}\) where \(a\) and \(b\) are the vector components along the x and y axes, respectively.
In our exercise, after calculating the vector components, we have the x-component as 9 and the y-component as 3.
Hence, the position vector \(\textbf{v}\) is written as \(9 \textbf{i} + 3 \textbf{j}\).
This standardized form makes it easy to understand and work with vectors in coordinate geometry and other branches of mathematics and physics.