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

Parametric equations: \(x(t) = -1 + 2t\), \(y(t) = 4 + 3t\).

Step 1: Understand the components of the problem

We are given a point (-1, 4)and a direction vector\[\begin{bmatrix}2 \3\end{bmatrix}\].The goal is to find the parametric equations of the line passing through the given point and following the direction of the vector.

Step 2: Write the general form of the parametric equation

A line in two-dimensional space can be represented by the parametric equations\(x = x_0 + at\)and\(y = y_0 + bt\), where \((x_0, y_0)\)is a point on the line,\( a\)and \(b\) are the components of the direction vector, and \(t\)is the parameter.

Step 3: Substitute the given point into the parametric equation

Use the given point (-1, 4)as \\((x_0, y_0))\):\(x = -1 + 2t\)\(y = 4 + 3t\).

Step 4: Write the final parametric equations

Therefore, the parametric equations of the line are:\(x(t) = -1 + 2t\)and\(y(t) = 4 + 3t\)

Key Concepts

Direction Vector

In the realm of parametric equations and lines, the direction vector plays a crucial role. It essentially determines the path that the line will take through space. Imagine you have a little arrow that points a specific way. That's what a direction vector does; it points the way your line will extend. In our exercise, the direction vector is \( \begin{bmatrix} 2 \ 3 \end{bmatrix} \). This means each step the line takes in the direction of the vector will move 2 units in the x-direction and 3 units in the y-direction. To construct a parametric equation, we attach this direction influence to a point.

• The direction vector provides the slope of the line.
• Each component of the vector demonstrates how far we move along each axis for every unit change in the parameter \(t\).
• This makes the line's path predictable and easy to follow.

Two-Dimensional Space

Two-dimensional space, commonly called the plane, is something we deal with every day, even if we don’t think about it. It's simply a flat surface that stretches endlessly in every direction.When dealing with points and lines in this space, we use coordinates to determine locations along the x and y axes. Each point on this plane is a combination of x and y values. In the current exercise, the point \((-1, 4)\) is such a point.

• The x-component tells us how far left or right the point is from the origin.
• The y-component tells us how far up or down the point is.

These components allow us to express any point or line as a combination of movements along these axes, making geometry both practical and navigable.

Lines in Geometry

Lines in geometry are fundamental concepts that seem simple but are quite profound. In essence, a line is a straight one-dimensional figure that extends indefinitely in both directions. It's defined by any two points or a point and a direction vector.In our exercise, the line is defined to pass through the point \((-1, 4)\) with a direction determined by the vector \( \begin{bmatrix} 2 \ 3 \end{bmatrix} \). To describe this line in terms of parametric equations:

• The x-component: \( x = -1 + 2t \)
• The y-component: \( y = 4 + 3t \)

These equations mean, starting at the point \((-1, 4)\), the line moves right 2 units and up 3 units for every unit increase in \(t\). The equations beautifully capture the infinite nature of lines by using the parameter \(t\), which can take any real number value, representing the entire line.