Problem 28 - 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 \(a \mathbf{i}+b \mathbf{j} .\) $$ P=(0,0) ; \quad Q=(-3,-5) $$

Step-by-Step Solution

Answer

\(\mathbf{v} = -3\mathbf{i} - 5\mathbf{j} \)

Step 1 - Identify the coordinates of initial and terminal points

The initial point is given as \( P = (0, 0) \). The terminal point is given as \( Q = (-3, -5) \).

Step 2 - Calculate the components of the vector

Find the difference between the corresponding coordinates of the terminal point and the initial point. The x-component is \( -3 - 0 = -3 \). The y-component is \( -5 - 0 = -5 \).

Step 3 - Formulate the position vector

Using the components calculated, the position vector can be written as: \[ \mathbf{v} = -3\mathbf{i} - 5\mathbf{j} \]

Key Concepts

Vectors in Algebra

Vectors are a fundamental concept in algebra. They are used to represent quantities that have both magnitude and direction.
A vector can be thought of as an arrow pointing from one point to another in space. The initial point of the vector is where the arrow starts, and the terminal point is where it ends.
To describe a vector accurately, we often break it down into its components. In two-dimensional space, these components are along the x-axis and the y-axis.
For example, the vector \(\textbf{v}\) with initial point \( P = (0,0) \) and terminal point \( Q = (-3,-5) \), can be broken down to find its x and y components.
The process involves subtracting the coordinates of the initial point from the coordinates of the terminal point to get the vector's components.

Coordinate Geometry

Coordinate geometry is the study of geometry using a coordinate system. This method allows for the precise placement and measurement of geometric figures in a plane. By using coordinates, you can easily transition from geometric concepts to algebraic expressions.
The position vector in coordinate geometry is expressed in terms of its components along the coordinate axes. For example, if the coordinates of initial point P are \( (x_1, y_1) \) and the coordinates of terminal point Q are \( (x_2, y_2) \), then the position vector \(\textbf{v}\) from P to Q can be calculated as:
$$ \textbf{v} = (x_2 - x_1) \mathbf{i} + (y_2 - y_1) \mathbf{j} $$
For our exercise, with P = (0,0) and Q = (-3,-5), the position vector is:
\[ \textbf{v} = (-3 - 0)\textbf{i} + (-5 - 0)\textbf{j} \]
This simplifies to:
\[ \textbf{v} = -3\textbf{i} - 5\textbf{j} \]

Vector Components

Vector components are the projections of a vector along the coordinate axes. In two-dimensional space, a vector can be broken down into horizontal (x) and vertical (y) components.
To find these components, you subtract the initial point coordinates from the terminal point coordinates.
Here is a step-by-step breakdown:

• Identify the coordinates of the initial point, P = (0,0).
• Identify the coordinates of the terminal point, Q = (-3,-5).
• Calculate the x-component: -3 - 0 = -3.
• Calculate the y-component: -5 - 0 = -5.

These components are then used to express the vector in component form:
\[ \textbf{v} = -3\textbf{i} - 5\textbf{j} \]
Each component tells you how far and in what direction the vector moves along each axis.