Problem 46 - Calculus for Biology and Medicine

Question

Find the parametric equation of the line in the \(x-y\) plane that goes through the indicated point in the direction of the indicated vector. $$(-1,4),\left[\begin{array}{l}2 \\ 3\end{array}\right]$$

Step-by-Step Solution

Answer

The parametric equations are: x = -1 + 2t, y = 4 + 3t.

Step 1: Identify Point and Direction Vector

Identify the given point through which the line passes, which is \((-1, 4)\), and the direction vector of the line, \(\begin{bmatrix} 2 \ 3 \end{bmatrix}\). This vector will help us determine the direction of the line.

Step 2: Write the Parametric Equations

The parametric equations of a line in the \(x-y\) plane can be written as \[x = x_0 + at \y = y_0 + bt\]where \((x_0, y_0)\) is the given point and \((a, b)\) are the components of the direction vector. Substituting in the values we have: \[x = -1 + 2t \y = 4 + 3t\]

Step 3: Review the Parametric Form

Double-check the parametric equations x = -1 + 2t y = 4 + 3t to ensure no errors in substituting the point and vector. Each equation should correctly represent the behavior of the line through \((-1,4)\) with the direction given by \([2,3]\).

Key Concepts

Parametric Form

A parametric form is a way of representing a line using parameters. Instead of expressing lines in the usual Cartesian format such as \(y = mx + b\), parametric equations use a third variable, often denoted as \(t\), to express coordinates \((x, y)\).

A line in the plane can be described by parametric equations:

• \(x = x_0 + at\)
• \(y = y_0 + bt\)

Here, \((x_0, y_0)\) represents a point on the line, and \(a\) and \(b\) are components of the direction vector, which allow the line to extend in both directions as \(t\) varies.

The parameter \(t\) is typically a real number, allowing you to trace the full extent of the line by adjusting \(t\). This form is very flexible and widely used in vector calculus and computer graphics.

Direction Vector

The direction vector is a crucial concept in parametric equations as it defines the line's orientation in the plane. In our exercise, the direction vector is given by \(\begin{bmatrix} 2 \ 3 \end{bmatrix}\).

When a line is defined parametrically, its direction vector provides the change in the \(x\) and \(y\) coordinates as the parameter \(t\) changes:

• The first component \(2\) signifies how much the \(x\)-coordinate increases or decreases for every unit increase in \(t\).
• The second component \(3\) indicates the corresponding change in the \(y\)-coordinate.

By scaling the direction vector with different values of \(t\), one can extend the line along its path, fully capturing its directional feature. This vector helps in quickly identifying the slope and direction of the line.

Line Equations

Parametric line equations represent lines in terms of parameter \(t\), offering a more versatile alternative to classical line equations. In our specific exercise, the line's equations from parametric form, derived from the point \((-1,4)\) and direction vector \(\begin{bmatrix} 2 \ 3 \end{bmatrix}\), are:

• \(x = -1 + 2t\)
• \(y = 4 + 3t\)

These equations provide an explicit formula for both \(x\) and \(y\) in terms of \(t\). By choosing any value for \(t\), you can calculate a corresponding point \((x, y)\) on the line.

Line equations in parametric form are particularly useful for problems involving motion along a line, intersections between lines, or generating graphical representations. They simplify the process of handling complex line interactions and extend the applicability of linear analysis beyond traditional forms.