Problem 12 - Advanced Engineering Mathematics

Question

In Problems, find parametric equations for the line through the given points. $$ (-3,7,9),(4,-8,-1) $$

Step-by-Step Solution

Answer

The parametric equations are \(x = -3 + 7t\), \(y = 7 - 15t\), \(z = 9 - 10t\).

Step 1: Identify the Points and Find the Direction Vector

We need to find the direction vector of the line. This is done by subtracting the coordinates of the first point, \((-3, 7, 9)\), from the second point, \((4, -8, -1)\). The direction vector \(\mathbf{v}\) is calculated as follows: \[ \mathbf{v} = (4 - (-3), -8 - 7, -1 - 9) = (7, -15, -10). \]

Step 2: Write Parametric Equations for the Line

Using the direction vector \((7, -15, -10)\) and the point \((-3, 7, 9)\), we can write the parametric equations for the line. For a line in three-dimensional space passing through a point \((x_0, y_0, z_0)\) with direction vector \((a, b, c)\), the parametric equations are: \[ x = x_0 + at, \quad y = y_0 + bt, \quad z = z_0 + ct. \]Substituting the point \((-3, 7, 9)\) and direction vector \((7, -15, -10)\) gives: \[ x = -3 + 7t, \quad y = 7 - 15t, \quad z = 9 - 10t. \]

Key Concepts

Direction Vector

When finding the equation of a line in three-dimensional space, one of the key components is the direction vector. This vector describes the line's orientation and indicates the direction in which the line extends.

To obtain the direction vector for a line passing through two points, say \(A(x_1, y_1, z_1)\) and \(B(x_2, y_2, z_2)\), you subtract coordinates component-wise. The result is a vector \( \mathbf{v} = (x_2 - x_1, y_2 - y_1, z_2 - z_1) \).

For instance, with points \( (-3, 7, 9) \) and \( (4, -8, -1) \), the calculation is:

• **X-component**: \(4 - (-3) = 7\)
• **Y-component**: \(-8 - 7 = -15\)
• **Z-component**: \(-1 - 9 = -10\)

This results in the direction vector \( \mathbf{v} = (7, -15, -10) \). This vector essentially tells us how the line progresses in each spatial dimension.

Three-Dimensional Space

Three-dimensional space can be quite fascinating with its depth and complexity. Unlike two-dimensional space, which only allows for movement left-right or up-down on a flat plane, three-dimensional space introduces a third axis, usually referred to as the z-axis.

In math, every point in this space is represented as \( (x, y, z) \) where:

• **x** is the position along the horizontal axis;
• **y** is the position along the vertical axis; and
• **z** is the position along the depth axis, which adds that additional dimension.

Three-dimensional geometry is used not only in mathematics but also in various practical applications like computer graphics, engineering, and physics. Here, the coordinate system allows us to describe shapes and lines in the space very precisely using equations, vector forms, or geometric formulas.

Understanding how objects are oriented in this space - such as the line created by two points - relies heavily on vectors and coordinate systems.

Lines in Geometry

Lines are fundamental in geometry, serving as the simplest form of curves in any space. In three-dimensional geometry, lines are characterized by parametric equations that use both a point on the line and a direction vector.

The parametric form is given as:\[\begin{align*} x &= x_0 + at, \ y &= y_0 + bt, \ z &= z_0 + ct\end{align*}\] Here, \( (x_0, y_0, z_0) \) represents a specific point on the line, and \( (a, b, c) \) is the direction vector. The parameter \( t \) can be thought of as a time variable that allows the point to "move" along the line as it changes.

• **At \( t = 0 \)**: The point is exactly \( (x_0, y_0, z_0) \).
• **As \( t \) increases or decreases**: The position shifts along the line influenced by the direction vector.

This provides the equations needed to express a line in a very comprehensive way, capturing all essential aspects of its infinite nature through tangible coordinates and vectors.