Problem 13 - Precalculus Mathematics for Calculus

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: Find the Direction Vector

To find the parametric equations of a line passing through two points, we first need the direction vector of the line. The direction vector \( \mathbf{v} \) can be found by subtracting the coordinates of point \( P \) from point \( Q \). \( \mathbf{v} = Q - P = (7-3, 3-7, -5+5) \) Thus, the direction vector \( \mathbf{v} = (4, -4, 0) \).

Step 2: Write the Parametric Equations

With the direction vector \( \mathbf{v} = (4, -4, 0) \) and a point on the line, such as \( P(3, 7, -5) \), we can write the parametric equations of the line. The parametric form of a line is given by:\[ x = x_0 + at \] \[ y = y_0 + bt \] \[ z = z_0 + ct \] where \((x_0, y_0, z_0)\) is a point on the line and \((a, b, c)\) is the direction vector. Substituting the values, we get:\[ x = 3 + 4t \] \[ y = 7 - 4t \] \[ z = -5 + 0t \] Thus, the parametric equations are:\( x = 3 + 4t \), \( y = 7 - 4t \), \( z = -5 \).

Step 3: Interpret the Parametric Equations

Each parametric equation represents how each coordinate of any point on the line changes with respect to the parameter \( t \). As \( t \) changes, it moves along the line defined by the direction vector, starting from the point \( P \). The equation \( z = -5 \) remains constant along the line, indicating that the line lies entirely in the plane where \( z = -5 \).

Key Concepts

Direction Vector

When trying to find the equation of a line in three-dimensional space, one of the key components we need is the direction vector. The direction vector is essentially a vector that points, or "directs," along the line.

To determine this direction vector from two points, we subtract the coordinates of these points. If you have two points on a line, let's say point \( P(x_1, y_1, z_1) \) and point \( Q(x_2, y_2, z_2) \), the direction vector \( \mathbf{v} \) can be found by the following formula:

• \( \mathbf{v} = Q - P = (x_2 - x_1, y_2 - y_1, z_2 - z_1) \)

The result \( \mathbf{v} \) provides the direction in which the line extends. This vector does not depend on the location of the points, just the direction. For our specific example, this direction vector is \( (4, -4, 0) \), meaning the line moves 4 units in the x-direction, -4 units in the y-direction, and remains constant in the z-direction.

Points on a Line

Any line in three-dimensional space can be uniquely determined by either two points that lie on the line or a point along with a direction vector.

When we look at the problem at hand, we are provided two points \( P(3, 7, -5) \) and \( Q(7, 3, -5) \). Each of these points is a position vector that essentially tells us the exact location of this point in 3D space. Once you know these coordinates, you can use them to express the line equation by determining which point you want your line to start from and adding a multiple of the direction vector to reach any other point on the line.

• Example starting point: \( \mathbf{r_0} = (3, 7, -5) \)

By applying the parametric equations, we incorporate these coordinates to traverse the line. Here, any point on the line can be expressed as \( \mathbf{r}(t) = \mathbf{r_0} + t\mathbf{v} \), where \( t \) is a parameter that moves the point along the line, adjusting according to the direction vector values.

Three-Dimensional Geometry

Three-dimensional geometry often involves analyzing points, lines, and surfaces within a space defined by three mutually perpendicular axes, labeled usually as x, y, and z.

Lines in 3D are significant because they demonstrate how an object can extend in multiple directions. Understanding the parametric form of such lines helps in visualizing and solving complex problems that involve spatial relationships.

• The x, y, and z components tell us how a point moves along the 3D line.
• For the exercise, the line is described by: \( x = 3 + 4t \), \( y = 7 - 4t \), and \( z = -5 \), where \( z = -5 \) indicates that the line remains in the plane \( z = -5 \).

This tracks that movement happens in the x and y directions, but remains consistent in the z direction, showcasing a horizontal plane line. Lines like these, when intersecting or existing in the same plane, are crucial to understanding 3D shapes like planes and surfaces.