Problem 43 - 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. $$(1,-1),\left[\begin{array}{l}2 \\ 1\end{array}\right]$$

Step-by-Step Solution

Answer

The parametric equations are \( x(t) = 1 + 2t \) and \( y(t) = -1 + t \).

Step 1: Understand the Concept of Parametric Equations for Lines

A parametric equation of a line in the plane is expressed as a set of equations that describe the position of a point on the line for each value of a parameter, typically represented by \( t \). The general form is \( x = x_0 + at \) and \( y = y_0 + bt \), where \((x_0, y_0)\) is a point on the line and \( (a, b) \) is the direction vector.

Step 2: Identify Given Point and Direction Vector

The given point is \((1,-1)\), which means \(x_0 = 1\) and \(y_0 = -1\). The direction vector is \( \begin{pmatrix} 2 \ 1 \end{pmatrix} \), which means \(a = 2\) and \(b = 1\). These values will be used to create the parametric equations.

Step 3: Write the Parametric Equations

Substitute the values into the general form. For the x-component: \( x = x_0 + at = 1 + 2t \). For the y-component: \( y = y_0 + bt = -1 + 1t = -1 + t \). Thus, the parametric equations of the line are: \[ x(t) = 1 + 2t \] \[ y(t) = -1 + t \]

Key Concepts

Parametric Equations of a Line

Parametric equations are a way to describe a line in coordinate geometry by expressing the coordinates of points on the line as functions of a parameter, often denoted by \( t \). Instead of the traditional slope-intercept or point-slope forms, parametric equations provide a more dynamic way to represent lines that is particularly useful in vector calculus and physics.

The general form of parametric equations for a line is given as:
\[ x(t) = x_0 + at \]
\[ y(t) = y_0 + bt \]
Here:

• \((x_0, y_0)\) represents a specific point through which the line passes, known as the position vector.
• \((a, b)\) is the direction vector that influences the direction and steepness of the line.

Using these equations, each value of \( t \) corresponds to a particular point on the line. Varying \( t \) traces out the entire line, making this form very versatile for various applications.

Direction Vector

The direction vector, denoted as \((a, b)\), plays a crucial role in defining the parametric equations of a line. This vector essentially gives the direction in which the line extends from the given point \((x_0, y_0)\).

For instance, a direction vector \( \begin{pmatrix} 2 \ 1 \end{pmatrix} \) implies that:

• For every increase of 1 unit in \( t \), the \( x \)-coordinate increases by 2 units.
• The \( y \)-coordinate increases by 1 unit.

Direction vectors can be thought of as guiding arrows along the path of the line. They also indicate how steep the line is and in which direction it moves in the coordinate plane.

Choosing different direction vectors from the same point produces different lines, highlighting the importance of this component in determining the characteristics of the line.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, blends algebra and geometry to offer a powerful method for studying geometric problems using a coordinate system. The revolution of this field lies in using the Cartesian plane—defined by the \( x \) and \( y \) axes—to represent geometric figures with equations.

Key features of coordinate geometry include:

• Ability to find distances between points using the distance formula.
• Calculating the slope of a line to understand its steepness and direction.
• Deriving the equations of lines, circles, and more complex curves.

When it comes to lines described through parametric equations, coordinate geometry provides a hands-on way to visualize these lines in a coordinate plane, making abstract algebraic concepts visually comprehensible. This geometrical perspective is invaluable for solving real-world applications in physics, engineering, and computer graphics.