Problem 29 - 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 \(a \mathbf{i}+b \mathbf{j} .\) $$ P=(3,2) ; \quad Q=(5,6) $$

Step-by-Step Solution

Answer

\(\mathbf{v} = 2\mathbf{i} + 4\mathbf{j}\).

Step 1: Identify Coordinates

Identify the coordinates of the initial point \(P\) and the terminal point \(Q\). \(P = (3, 2)\) and \(Q = (5, 6)\).

Step 2: Calculate the Vector Components

Compute the components of the vector \(\mathbf{v}\) by subtracting the coordinates of \(P\) from \(Q\). The component form is calculated as \( (x_2 - x_1, y_2 - y_1)\): \( (5 - 3, 6 - 2)\).

Step 3: Simplify the Components

Simplify the subtraction to find the components of the vector: \( (2, 4)\).

Step 4: Express in Vector Form

Express the vector \(\mathbf{v}\) in the form \(a \mathbf{i} + b \mathbf{j} \). Using the components \(a = 2 \) and \(b = 4\): \(\mathbf{v} = 2 \mathbf{i} + 4 \mathbf{j}\).

Key Concepts

Position Vector

In vector mathematics, a position vector defines the position of a point in a space relative to an origin. Imagine you are standing on a map, and you want to describe to someone where a specific location is.
Instead of giving directions, you give them coordinates.
For instance, if we want to find the position vector of a point, we need the coordinates of that point.
In our exercise, the position vector we are talking about is between points \(P\) and \(Q\).
This vector, \(\textbf{v}\), starts from point \(P\) (initial point) and points to \(Q\) (terminal point).

Subtraction of Coordinates

To find the components of a vector, you subtract the coordinates of the initial point from the coordinates of the terminal point.
Let's break down our example: We have the points \(P = (3, 2)\) and \(Q = (5, 6)\).
In general, if you have two points, \(P = (x_1, y_1)\) and \(Q = (x_2, y_2)\), you find the vector components by calculating \((x_2 - x_1, y_2 - y_1)\).
In our exercise:

• Subtract \(3\) from \(5\) to get the \(x\)-component: \(5 - 3 = 2\).
• Subtract \(2\) from \(6\) to get the \(y\)-component: \(6 - 2 = 4\).

This gives us the vector components \((2, 4)\).

Vector Components

Vector components represent the movement from the initial point to the terminal point in each dimension (or direction).
In our 2D space example, the vector \(\textbf{v}\) is \((2, 4)\), where:

• The \(2\) in \(2\textbf{i}\) represents 2 units moved along the \(x\)-axis.
• The \(4\) in \(4\textbf{j}\) represents 4 units moved along the \(y\)-axis.

Essentially, vector components break down the overall movement into movements along each axis.
This makes it easier to understand and work with vectors in different dimensions.

Vector Notation

Vectors can be expressed in various notations to suit different contexts.
In our exercise, we express the vector \(\textbf{v}\) in the form \(a\textbf{i} + b\textbf{j}\):

• \(\textbf{i}\) represents the unit vector in the \(x\)-direction (horizontal).
• \(\textbf{j}\) represents the unit vector in the \(y\)-direction (vertical).
• The coefficients \(a\) and \(b\) show how much to scale the unit vectors \(\textbf{i}\) and \(\textbf{j}\).

For \(\textbf{v}\), where \(a = 2\) and \(b = 4\), the vector notation is:\textbf{v} = 2\textbf{i} + 4\textbf{j}
This notation not only clearly depicts the components but also visually represents the direction and magnitude of the vector.