Problem 32 - Calculus 3

Question

The line through \(P\) and \(Q\) where \(P\) is \((1,4,-2)\) and \(Q\) is \((3,9,6)\)

Step-by-Step Solution

Answer

The parametric equations for the line are: \(x = 1 + 2t\), \(y = 4 + 5t\), \(z = -2 + 8t\).

Step 1: Determine the direction vector

To find the line through points \(P\) and \(Q\), first determine the direction vector, \(\mathbf{d}\), by subtracting the coordinates of \(P\) from \(Q\): \(\mathbf{d} = (3-1, 9-4, 6-(-2))\). This simplifies to \(\mathbf{d} = (2, 5, 8)\).

Step 2: Write the parametric equations

Using the direction vector \(\mathbf{d} = (2, 5, 8)\) and point \(P = (1, 4, -2)\), we can write the parametric equations of the line. Let \(t\) be the parameter, then the equations are: \(x = 1 + 2t\), \(y = 4 + 5t\), and \(z = -2 + 8t\).

Step 3: Express the vector equation

The vector equation of the line can be expressed as: \(\mathbf{r}(t) = (1, 4, -2) + t(2, 5, 8)\). This shows the line as a function of \(t\), where \(\mathbf{r}(t)\) represents any point on the line.

Key Concepts

Direction Vector

In vector calculus, the direction vector is an essential concept. It helps define the path of a line in three-dimensional space. To find a direction vector between two points, like in the provided exercise with points \( P(1,4,-2) \) and \( Q(3,9,6) \), you calculate the difference between each corresponding coordinate of the two points. This gives you \( \mathbf{d} = (3-1, 9-4, 6-(-2)) = (2,5,8) \). This vector, \( \mathbf{d} \), indicates the direction from point \( P \) to point \( Q \).

Why is this important? The direction vector lets you express the line algebraically. Having a clear sense of direction is crucial for both graphical representations and algebraic descriptions.

Key Points:

• Direction vectors are created by subtracting the coordinates of the starting point from the ending point.
• They are vital for determining the orientation of a line in space.

Parametric Equations

Parametric equations are a powerful way to represent a line using a parameter, typically denoted by \( t \). For our exercise, we use the point \( P(1,4,-2) \) and the direction vector \( \mathbf{d} = (2,5,8) \) to define the line's parametric equations as: \( x = 1 + 2t \), \( y = 4 + 5t \), and \( z = -2 + 8t \). Here, \( t \) can be any real number, which adjusts which specific point along the line you are examining.

These equations are particularly helpful because they allow for an intuitive understanding of the line. By changing \( t \), you can quickly determine the coordinates of various points on the line.

Benefits of Parametric Equations:

• They provide flexibility in describing curves and lines in terms of position.
• They reveal how each dimension changes with respect to the parameter \( t \).

Vector Equation

The vector equation of a line is another robust way to describe a line in space, emphasizing its geometric properties. For the given line through points \( P(1,4,-2) \) and with direction vector \( \mathbf{d} = (2,5,8) \), the vector equation is written as: \[ \mathbf{r}(t) = (1, 4, -2) + t(2, 5, 8) \].

This equation states that any point on the line, \( \mathbf{r}(t) \), is the sum of the initial point \( P \) and a scalar multiple \( t \) of the direction vector.

Why use a vector equation? It provides a clear and concise representation that combines both the position (through the starting point) and direction (through the direction vector) of the line.

Takeaways from a Vector Equation:

• It visually and mathematically describes how to traverse the line from an initial point using the direction vector.
• Useful for calculating the position of any point along the line intuitively by varying \( t \).