Problem 27 - 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 \(a \mathbf{i}+b \mathbf{j} .\) $$ P=(0,0) ; \quad Q=(3,4) $$

Step-by-Step Solution

Answer

\mathbf{v} = 3 \mathbf{i} + 4 \mathbf{j}

Step 1: Understand the Problem

Given the initial point P and the terminal point Q, the aim is to find the position vector \mathbf{v} in the form \(a \mathbf{i} + b \mathbf{j} \).

Step 2: Identify Coordinates of Points

The coordinates of point P are given as \(0,0\) and the coordinates of point Q are given as \(3,4\).

Step 3: Calculate the Components of the Vector

To find the components of the vector \mathbf{v}, subtract the coordinates of P from the coordinates of Q. \mathbf{v} = Q - P = \(3 - 0, 4 - 0\) = \(3,4\).

Step 4: Express the Position Vector

Express the position vector \mathbf{v} in the form \(a \mathbf{i} + b \mathbf{j}\). Since \(a = 3\) and \(b = 4\), \mathbf{v} can be written as: \mathbf{v} = 3 \mathbf{i} + 4 \mathbf{j}.

Key Concepts

Vector Components

Vectors have two important components that help describe their direction and magnitude. These components are the differences in the x-coordinates and y-coordinates between the initial and terminal points. For example, if we have an initial point P at (0,0) and a terminal point Q at (3,4), the components of the vector \( \mathbf{v} \) are:

• The x-component: Difference in x-coordinates, calculated as \[ 3 - 0 = 3 \]
• The y-component: Difference in y-coordinates, calculated as \[ 4 - 0 = 4 \]

Together, these components form the vector \[ \mathbf{v} = (3, 4) \]. These are essentially the 'building blocks' of the vector, specifying how far it moves horizontally and vertically.

Vector Notation

Vector notation is how we write down vectors so that we know their directions and magnitudes. A common way to express a vector is in the form \[ a \mathbf{i} + b \mathbf{j} \], where:

• \[ a \] represents the x-component
• \[ b \] represents the y-component

The letters \( \mathbf{i} \) and \( \mathbf{j} \) are unit vectors, meaning that they have a magnitude of 1 and point in the directions of the x and y axes, respectively. So, for our position vector with components (3,4), we can write it as: \[ \mathbf{v} = 3 \mathbf{i} + 4 \mathbf{j} \]. This notation gives us a clear and concise way to understand the direction and magnitude of the vector.

Coordinate Points

Coordinate points, like the initial point P and terminal point Q, are crucial for defining vectors. Points are defined by their coordinates \( (x, y) \) in a two-dimensional space. These coordinates tell us where the points are located. For example:

• The point P is given as \( (0,0) \), which is the origin of the coordinate plane.
• The point Q is given as \( (3,4) \), telling us the vector travels 3 units to the right and 4 units up from the origin.

By knowing these coordinates, we can easily find the components of the vector by calculating the differences in x and y values between points P and Q, leading us to our position vector.