Problem 17 - Advanced Engineering Mathematics

Question

In Problems, find symmetric equations for the line through the given points. $$ (5,10,-2),(5,1,-14) $$

Step-by-Step Solution

Answer

The symmetric equations are \(x = 5, \frac{y - 10}{-9} = \frac{z + 2}{-12}\).

Step 1: Identify Given Points

The two points given are \((5, 10, -2)\) and \((5, 1, -14)\). These will be used to find the direction vector of the line.

Step 2: Compute the Direction Vector

To find the direction vector \(\mathbf{d}\), subtract the coordinates of the first point from the second: \[ \mathbf{d} = (5-5, 1-10, -14-(-2)) = (0, -9, -12). \]

Step 3: Write the Parametric Equations

The parametric equations for a line through a point \((x_0, y_0, z_0)\) with direction vector \((a, b, c)\) are: \[ x = x_0 + at, \quad y = y_0 + bt, \quad z = z_0 + ct. \] Use the point \((5, 10, -2)\) and direction vector \((0, -9, -12)\) to write: \[ x = 5, \quad y = 10 - 9t, \quad z = -2 - 12t. \]

Step 4: Write the Symmetric Equations

Since \(x\) does not change, its symmetric equation is simply \(x = 5\). For the symmetric form, solve the parametric equations for \(t\): \[ x = 5, \quad \frac{y - 10}{-9} = \frac{z + 2}{-12}. \]Here the symmetric equations rely on the constant \(x = 5\), hence only two equations are needed.

Key Concepts

Direction Vector

In three-dimensional space, a direction vector helps us determine the path or direction of a line. This vector is essential when calculating the way a line extends through the points. To find the direction vector between two points, you subtract the coordinates of one point from the other.

For instance, given the points \((5, 10, -2)\) and \((5, 1, -14)\), the direction vector \(\mathbf{d}\) is found as \((5-5, 1-10, -14-(-2))\). This simplifies to \((0, -9, -12)\). Each element of this vector points in the direction from the first point to the second.

• The first component \(0\) means no movement along the x-axis.
• The second component \(-9\) indicates a downward movement on the y-axis.
• The last component \(-12\) signifies downward movement along the z-axis.

This vector becomes crucial when you want to express the line as an equation in either a parametric or symmetric form.

Parametric Equations

Parametric equations of a line are a way to describe a line using a set of parameters. Instead of relying solely on coordinate points, these equations use a parameter (usually \(t\)) to specify coordinates.

Given a starting point \((x_0, y_0, z_0)\) and a direction vector \((a, b, c)\), the parametric equations are:

• \(x = x_0 + at\)
• \(y = y_0 + bt\)
• \(z = z_0 + ct\)

These equations express the x, y, and z coordinates as separate functions of \(t\), showing where the line moves as \(t\) changes.

Using the point \((5, 10, -2)\) and direction vector \((0, -9, -12)\), we get the parametric equations as:

• \(x = 5\)
• \(y = 10 - 9t\)
• \(z = -2 - 12t\)

The beauty of parametric equations is their simplicity in mapping the entire line, as moving the parameter \(t\) will continuously produce points along the line.

Points in Space

Points in space are simply a set of coordinates \((x, y, z)\) that determine a location in three-dimensional space. When you're given two points, like \((5, 10, -2)\) and \((5, 1, -14)\), they serve as anchors from which lines and planes can be derived or defined.

Understanding the concept of points is essential when exploring relationships such as distances or the equation of a line formed by these points. Each coordinate represents a distance from a defined origin along the respective axis:

• \(x\) coordinate determines how far to the right or left a point is from the y-z plane.
• \(y\) coordinate indicates how far in front or behind a point is from the x-z plane.
• \(z\) coordinate shows how high or low a point is from the x-y plane.

By using these coordinates, you can construct lines like in the exercise, find direction vectors for defining motion or direction, and derive equations that help in understanding spatial relationships between points.