Problem 33 - 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 \(a \mathbf{i}+b \mathbf{j} .\) $$ P=(1,0) ; \quad Q=(0,1) $$

Step-by-Step Solution

Answer

\(\textbf{v} = - \textbf{i} + \textbf{j}\)

Step 1: Understand the points

The vector \(\textbf{v}\) has initial point \(P = (1,0)\) and terminal point \(Q = (0,1)\). We need to find the position vector.

Step 2: Calculate the components

The components of the position vector \(\textbf{v}\) are found by subtracting the corresponding components of point \(P\) from point \(Q\). Thus, the x-component \(v_x\) is \(0 - 1 = -1\) and the y-component \(v_y\) is \(1 - 0 = 1\).

Step 3: Express the position vector

Now, the position vector \(\textbf{v}\) can be written in the form \(a \textbf{i} + b \textbf{j}\) where \(a = -1\) and \(b = 1\). Therefore, \(\textbf{v} = - \textbf{i} + \textbf{j}\).

Key Concepts

Vector Components

Let’s break down the concept of vector components.
Vectors are represented in terms of their components along the axes of a coordinate system.
For a 2D vector, these are the x-component and the y-component.

For the given problem, the initial point is \(P = (1,0)\) and the terminal point is \(Q = (0,1)\).
The components of the vector \(\mathbf{v}\) can be found by subtracting the coordinates of point P from those of point Q.
Here’s how the calculation goes:

• The x-component: \(v_x = 0 - 1 = -1\)
• The y-component: \(v_y = 1 - 0 = 1\)

These components essentially describe how far and in what direction you need to travel along the x-axis and y-axis to get from point P to point Q.

Initial Point

The initial point of a vector is the point where the vector starts.
In this problem, that's point \(P = (1,0)\).

Understanding the initial point is crucial because it serves as a reference for determining the direction and length (magnitude) of the vector.
When working with vectors, always note the initial point coordinates. These coordinates are used in subtracting to find the vector components.

For instance, if point P was different, the components you get when performing the subtraction would also be different, leading to a different vector.

Terminal Point

The terminal point is where the vector ends.
In this example, the terminal point is \(Q = (0,1)\).

Much like the initial point, the terminal point is essential for finding the vector’s direction and length.
By subtracting the coordinates of the initial point from those of the terminal point, you accurately define how far and in what direction the vector points from its start.

Using the coordinates:

• Terminal point Q: \(Q = (0,1)\)

Without the correct terminal point coordinates, you'll end up with incorrect component values and an inaccurate representation of the vector.

Vector Notation

Vectors are often represented in notation that succinctly describes their direction and magnitude.
The most common form for in-plane vectors is \(a \mathbf{i} + b \mathbf{j}\), where:

• \(\mathbf{i}\) represents the unit vector in the x-direction
• \(\mathbf{j}\) represents the unit vector in the y-direction
• \(a\) and \(b\) are the vector’s components

In this exercise, after determining the components:

• The x-component is \(a = -1\)
• The y-component is \(b = 1\)

The vector \(\mathbf{v}\) finally is written as \(\mathbf{v} = - \mathbf{i} + \mathbf{j}\).
This notation clearly shows both the direction and magnitude in compact form.