Problem 13 - Algebra and Trigonometry

Question

Find parametric equations for the line that passes through the points \(P\) and \(Q .\) $$ P(3,7,-5), \quad Q(7,3,-5) $$

Step-by-Step Solution

Answer

The parametric equations are: \(x = 3 + 4t\), \(y = 7 - 4t\), \(z = -5\).

Step 1: Determine the Direction Vector

First, find the direction vector of the line by subtracting the coordinates of point \(P\) from point \(Q\). This vector can be found by computing \((7-3, 3-7, -5-(-5)) = (4, -4, 0)\).

Step 2: Write the Parametric Equations

Using point \(P(3,7,-5)\) and the direction vector \((4, -4, 0)\), write the parametric equations for the line: \[x = 3 + 4t, \quad y = 7 - 4t, \quad z = -5 + 0t \] Simplify to: \[x = 3 + 4t, \quad y = 7 - 4t, \quad z = -5 \]

Key Concepts

Direction Vector

In the world of parametric equations, understanding the direction vector is key when defining the path of a line. To find the direction vector, you need to know two points through which the line passes. Let's look at our example points, \( P(3, 7, -5) \) and \( Q(7, 3, -5) \). The direction vector is essentially a vector that points from one of these points to the other. You find it by subtracting the coordinates of point \(P\) from point \(Q\).

Here's how it works:

• Subtract the x-coordinates: \(7-3 = 4\)
• Subtract the y-coordinates: \(3-7 = -4\)
• Subtract the z-coordinates: \(-5 - (-5) = 0\)

This gives us the direction vector \((4, -4, 0)\). This vector shows the direction and rate of movement along the line in 3D space.
It indicates that for every step you take in the x-direction, you should step 4 units forward, 4 units back in y, and no movement in z.

Parametric Equations for a Line

Parametric equations allow you to express each coordinate of a line as a function of a single parameter, typically denoted as \(t\). This can beautifully capture the behavior of a line in three-dimensional space, describing how the x, y, and z coordinates change as you move along the line.

Using point \( P(3, 7, -5) \), our starting location, and our direction vector \((4, -4, 0)\), the parametric equations of the line are formed as follows:

• For x: \(x = 3 + 4t\)
• For y: \(y = 7 - 4t\)
• For z: \(z = -5 + 0t\)

These equations tell us exactly where \(x\), \(y\), and \(z\) will be for any given \(t\). When \(t = 0\), you find yourself at point \(P\).
As \(t\) increases or decreases, you move along the line, with \(t\) serving as an indicator of how far you've traveled.

Coordinates of Points

Coordinates are essential in describing the location of points in space. Any point in three dimensions will have coordinates in the form \((x, y, z)\). In our example, we start with points \( P(3, 7, -5) \) and \( Q(7, 3, -5) \).

To understand how the parametric equations define the line passing through these points, consider how coordinates function:

• The x-coordinate determines how far left or right you are.
• The y-coordinate determines how far backward or forward you are.
• The z-coordinate determines how high or low you are.

Through parametric equations, the coordinates of every point along the line can be derived.
For any value of \(t\), the coordinates \((x(t), y(t), z(t))\) offer the precise location on the line. This ability to pinpoint exact coordinates makes parametric equations a powerful tool in three-dimensional geometry.