Problem 13 - Advanced Engineering Mathematics

Question

In Problems, find symmetric equations for the line through the given points. $$ (1,4,-9),(10,14,-2) $$

Step-by-Step Solution

Answer

Symmetric equations: \( \frac{x-1}{9} = \frac{y-4}{10} = \frac{z+9}{7} \).

Step 1: Find the Direction Vector

To find the symmetric equations of a line through two points, we first need the direction vector. Subtract the coordinates of the first point from the coordinates of the second point to get the direction vector. If the points are \((x_1, y_1, z_1) = (1,4,-9)\) and \((x_2, y_2, z_2) = (10,14,-2)\), the direction vector is given by \((x_2-x_1, y_2-y_1, z_2-z_1)\). Calculate: \( (10-1, 14-4, -2+9) = (9, 10, 7) \).

Step 2: Write the Parametric Equations

Using the point \((1, 4, -9)\) and the direction vector \((9, 10, 7)\), write the parametric equations of the line. The parametric equations are: \(x = 1 + 9t\), \(y = 4 + 10t\), and \(z = -9 + 7t\), where \(t\) is a parameter.

Step 3: Derive the Symmetric Equations

Equate each parametric equation to isolate \(t\) as follows: \(x = 1 + 9t\) gives \(t = \frac{x-1}{9}\), \(y = 4 + 10t\) gives \(t = \frac{y-4}{10}\), and \(z = -9 + 7t\) gives \(t = \frac{z+9}{7}\). Combine these equalities to eliminate \(t\) and find the symmetric equations: \( \frac{x-1}{9} = \frac{y-4}{10} = \frac{z+9}{7} \).

Key Concepts

Direction Vector

In 3D coordinate geometry, a direction vector is the backbone of a line. It dictates the path or direction in which the line extends. To find the direction vector between two points, subtract the coordinates of the first point from those of the second point. Consider points

• First point: \((x_1, y_1, z_1) = (1, 4, -9)\)
• Second point: \((x_2, y_2, z_2) = (10, 14, -2)\)

To derive the direction vector \((d_1, d_2, d_3)\), calculate it as follows:\[d_1 = x_2 - x_1 = 10 - 1 = 9\ d_2 = y_2 - y_1 = 14 - 4 = 10 \ d_3 = z_2 - z_1 = -2 + 9 = 7\]Thus, the direction vector is \((9, 10, 7)\). This means the line moves 9 units in the x-direction, 10 units in the y-direction, and 7 units in the z-direction. Each segment of this direction guides the slope of the line in three-dimensional space.

Parametric Equations

Parametric equations express each coordinate of a point on a line in terms of a single parameter, typically denoted as \(t\). This is a powerful way to represent a line in 3D, allowing us to describe every point along it. Using the point \((1,4,-9)\) and the direction vector \((9,10,7)\), we set up the following parametric equations for the line:

• x-dimension: \(x = 1 + 9t\)
• y-dimension: \(y = 4 + 10t\)
• z-dimension: \(z = -9 + 7t\)

Here, \(t\) is a scalar parameter that varies over the entire real number line, allowing us to reach any point on the line. The structure of each parametric equation reveals how the line extends from the initial point. With these equations, any line in 3D space can be uniquely defined by a starting point and a direction vector.

3D Coordinate Geometry

3D coordinate geometry extends the ideas of 2D geometry to three dimensions. It involves the representation of points, lines, and surfaces in a space determined by three axes: x, y, and z. A key concept in three-dimensional geometry is the line, represented in different forms such as:

• Point-direction form through parametric equations
• Symmetric form, which involves eliminating the parameter \(t\)

In symmetric form, each parametric equation is set to solve for \(t\). For the given parametric equations:\[t = \frac{x-1}{9}, \quad t = \frac{y-4}{10}, \quad t = \frac{z+9}{7}\]Setting these equations equal to one another eliminates \(t\) and provides the symmetric equations:\[\frac{x-1}{9} = \frac{y-4}{10} = \frac{z+9}{7}\]This tells us that all points lying on the line satisfy this condition, offering another way to describe lines in our three-dimensional space effectively. Understanding 3D coordinate geometry equips one to handle complex spatial problems with ease.