Problem 5 - Calculus Early Transcendentals: Pearson New International Edition

Question

Write both the parametric equations and the symmetric equations for the line through the given point parallel to the given vector. \((4,5,6),\langle 3,2,1\rangle\)

Step-by-Step Solution

Answer

Parametric: \(x = 4 + 3t\), \(y = 5 + 2t\), \(z = 6 + t\); Symmetric: \(\frac{x-4}{3} = \frac{y-5}{2} = z-6\).

Step 1: Identify Given Point and Direction Vector

We are given a point \((4, 5, 6)\) and a direction vector \(\langle 3, 2, 1 \rangle\). We will use these to write the parametric and symmetric equations of the line.

Step 2: Write the Parametric Equations

The parametric equations of a line can be expressed as \(x = x_0 + at\), \(y = y_0 + bt\), \(z = z_0 + ct\), where \((x_0, y_0, z_0)\) is the point on the line and \(\langle a, b, c \rangle\) is the direction vector. Substituting the given point and vector, we have:\[ x = 4 + 3t \]\[ y = 5 + 2t \]\[ z = 6 + 1t \]

Step 3: Write the Symmetric Equations

Symmetric equations are derived by eliminating the parameter \(t\) from the parametric equations. Setting each parametric equation equal to \(t\), we have:\[ t = \frac{x - 4}{3} \]\[ t = \frac{y - 5}{2} \]\[ t = z - 6 \]Thus, the symmetric equations can be written as:\[ \frac{x - 4}{3} = \frac{y - 5}{2} = z - 6 \]

Key Concepts

Symmetric Equations

Symmetric equations provide a concise and elegant way to describe lines in three-dimensional space. They are useful because they eliminate the parameter, making the relation between the coordinates more direct. By setting each component of the parametric equation equal to the same parameter \(t\), we derive the symmetric form. For the line through the point \((4, 5, 6)\) with the direction vector \(\langle 3, 2, 1 \rangle\), the symmetric equations are

• \(\frac{x - 4}{3} = \frac{y - 5}{2} = z - 6\)

This form emphasizes that the ratio of change in position within each coordinate is consistent along the line. Symmetric equations offer a straightforward method to understand how each coordinate depends equally on one common parameter while traversing the line.

Direction Vector

A direction vector is crucial when defining a line, especially in three-dimensional coordinate geometry. It shows the direction in which the line extends. Given a direction vector \(\langle a, b, c \rangle\), it tells us how much to move in each coordinate direction to step along the line. In our example, the direction vector is \(\langle 3, 2, 1 \rangle\).

• The number 3 indicates movement along the x-axis.
• The number 2 shows the shift on the y-axis.
• The number 1 provides the change on the z-axis.

Together, these values influence how the line stretches and orients itself across the 3D space. The length of the direction vector isn’t as important as its direction, serving as a "guide" for the trajectory of the line.

Line Equation

The line equation in parametric form summarizes how we can reach any point on the line by varying the parameter \(t\). It uses a specific point on the line and a direction vector. In our parametric equations,

• \(x = 4 + 3t\)
• \(y = 5 + 2t\)
• \(z = 6 + 1t\)

We see that each coordinate equation adjusts linearly with \(t\). This parametric form is incredibly helpful because:

• It clearly indicates both a specific point on the line (the starting point)
• Shows how to extend along the line (using the direction vector)

By adjusting \(t\), you move along the line, with the initial point \((4, 5, 6)\) shifted by multiples of the direction vector \(\langle 3, 2, 1 \rangle\).

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, enables us to describe the geometric shapes using equations. By using numbers and equations, it offers a practical way to represent lines, curves, and other shapes in a coordinate system. In three-dimensional space, like in our example, it becomes crucial to understand how points, lines, and planes interact.

• Points are described by coordinates \((x, y, z)\).
• Lines use parametric and symmetric equations.
• Planes might be expressed with a certain equation \(Ax + By + Cz = D\).

Coordinate geometry simplifies the relations between these entities using algebraic equations. We precisely define spatial elements and predict how changes in variables alter their positions, playing a core role in fields ranging from mathematics to engineering and computer graphics.