Question

All lines are in the \((x, y)\) plane. Find the slope of the line whose parametric equation is \(\mathbf{r}=(\mathbf{i}-\mathbf{j})+(2 \mathbf{i}+3 \mathbf{j}) t\).

Short answer

The slope of the line is \( \frac{3}{2} \).

Step-by-step solution

- Understand the Parametric Equation

Given the parametric equation: \(\textbf{r} = (\textbf{i} - \textbf{j}) + (2\textbf{i} + 3\textbf{j})t\) This describes the position vector of any point on the line in terms of parameter \(t\).

- Identify the Direction Vector

The line's direction vector is the coefficient of \(t\). From \( \textbf{r} = (\textbf{i} - \textbf{j}) + (2\textbf{i} + 3\textbf{j})t\), it is \(2\textbf{i} + 3\textbf{j}\).

- Find Components of Direction Vector

The direction vector \(2\textbf{i} + 3\textbf{j}\) has components;\(a = 2\) (coefficient of \( \textbf{i}\)) and \(b = 3\) (coefficient of \( \textbf{j}\)).

- Calculate the Slope

The slope \(m\) of the line can be found using the formula: \(m = \frac{b}{a} = \frac{3}{2}\).

Key concepts

Parametric Equations

In mathematics, a parametric equation represents a curve as a set of equations. Instead of writing equations in the form of \(y = mx + c\), parameter equations use a parameter, typically denoted by \(t\).

For a line, the parametric equation can be represented as:
\[ \mathbf{r} = \textbf{r}_0 + \textbf{v} t \]

Here, \( \textbf{r}_0 \) is the position vector, \( \textbf{v} \) is the direction vector, and \( t \) is the parameter. This representation makes it easier to describe lines and curves in space by breaking them down into x and y components.

Let's break down the given parametric equation:
\[ \textbf{r} = ( \textbf{i} - \textbf{j} ) + (2 \textbf{i} + 3 \textbf{j})t \]

In this equation, \( \textbf{i} \) and \( \textbf{j} \) are unit vectors in the x and y directions respectively. In simple terms, this equation tells us the position of any point on the line as \( t \) changes.

Direction Vector

The direction vector of a line indicates the direction in which the line extends.

In a parametric equation like:
\[ \textbf{r} = ( \textbf{i} - \textbf{j} ) + (2 \textbf{i} + 3 \textbf{j})t \]
The direction vector is the coefficient of the parameter, \( t \).

Here, it is:
\[ 2 \textbf{i} + 3 \textbf{j} \]
This means the line is directed 2 units in the x-direction and 3 units in the y-direction. The direction vector helps us understand the line's steepness and direction in the coordinate plane.
One important thing to note is that the direction vector does not depend on the position vector, meaning the slope and direction of the line remain the same.

Vector Components

Breaking down vectors into their components makes them easier to work with. Any vector in a plane can be represented as a combination of unit vectors.

For example, in our direction vector:
\[ 2 \textbf{i} + 3 \textbf{j} \]
This means:

• 2 is the component in the direction of the x-axis (\( \textbf{i} \)).
• 3 is the component in the direction of the y-axis (\( \textbf{j} \)).

The direction vector \(2 \textbf{i} + 3 \textbf{j}\) thus consists of its components 2 and 3. This separation allows us to calculate the slope and the direction of the line accurately. Components are crucial when switching between different forms of equations or transforming geometric entities.

Calculating Slope

The slope of a line indicates its steepness and direction. For a line with a parametric equation, the slope can be found using the components of the direction vector.

Given our direction vector:
\[ 2 \textbf{i} + 3 \textbf{j} \]Break this down into components:

• The x-component \( a = 2 \).
• The y-component \( b = 3 \).

Use these components to find the slope \( m \). The formula for finding the slope from the direction vector is:
\[ m = \frac{b}{a} \]

Substitute the values of the components into the formula:
\[ m = \frac{3}{2} \]
The slope of this line is \( \frac{3}{2} \), meaning the line rises 3 units for every 2 units it runs. Understanding how to calculate the slope using direction vectors is essential for interpreting parametric equations and analyzing geometric properties of lines.