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 \(a \mathbf{i}+b \mathbf{j} .\) $$ P=(1,1) ; \quad Q=(2,2) $$

Short answer

\( \textbf{v} = 1 \textbf{i} + 1 \textbf{j} \)

Step-by-step solution

Understand the Problem

The task is to find the position vector \(\textbf{v} \) which starts at point \(\textbf{P} = (1, 1 )\) and ends at point \(\textbf{Q} = (2, 2) \). The position vector should be expressed in the form of \(\textbf{v} = a \textbf{i} + b \textbf{j}\).

Determine the Components of the Position Vector

The components of the vector \(\textbf{v}\) can be found by subtracting the coordinates of \(\textbf{P}\) from the coordinates of \(\textbf{Q}\). The horizontal component \((a)\) is the difference between the x-coordinates: \((2 - 1) = 1\). The vertical component \((b)\) is the difference between the y-coordinates: \((2 - 1) = 1\).

Write the Position Vector

Combine the components obtained in Step 2 to express the vector \(\textbf{v}\) in the required form. Therefore, \(\textbf{v} = 1 \textbf{i} + 1 \textbf{j}\).

Key concepts

Vector Components

When dealing with vectors in algebra, it's essential to understand their components. A vector is often represented in terms of its horizontal and vertical parts.
Think of a vector as an arrow pointing from one location to another.
The vector’s components break this arrow down into two parts: one that stretches horizontally and one that stretches vertically.
In the example given, the components of the vector \(\textbf{v}\) are found by subtracting the corresponding coordinates of its initial and terminal points.
For the horizontal component (a), you take the difference in the x-coordinates, and for the vertical component (b), you take the difference in the y-coordinates.
These components are what you use to describe the vector in a clear and manageable form.

Initial and Terminal Points

To understand a vector's journey, you need to know where it starts and where it ends.
The initial point is where the vector begins, while the terminal point is where it finishes.
In the provided exercise, the initial point \(P\) is given as \( (1, 1) \), and the terminal point \(Q\) is \( (2, 2) \).
By knowing these points, you can find the vector's position by looking at the difference between these coordinates.
Subtract the coordinates of the initial point from the terminal point to find the components of the position vector.
This lets you understand how far and in which direction the vector travels from start to finish.

Vector Notation

After finding the components of a vector, it's essential to write it in a clear and standard notation.
This makes it easier to read, interpret, and use in further calculations.
In our case, we express the vector \( \textbf{v} \) in the form \( a \textbf{i} + b \textbf{j} \).
Here, \( \textbf{i} \) and \( \textbf{j} \) are unit vectors along the horizontal (x) and vertical (y) axes, respectively.
The coefficients \a \ and \b \ come directly from the components we calculated.
In this exercise, both components are 1, so the vector is written as \( \textbf{v} = 1 \textbf{i} + 1 \textbf{j} \).
This notation not only gives the direction but also the magnitude of the vector in a clear and standardized way.