Question

Write both the parametric equations and the symmetric equations for the line through the given point parallel to the given vector. \((-2,2,-2),(7,-6,3)\)

Short answer

Parametric: \(x = -2 + 7t\), \(y = 2 - 6t\), \(z = -2 + 3t\). Symmetric: \(\frac{x+2}{7} = \frac{y-2}{-6} = \frac{z+2}{3}\).

Step-by-step solution

Identify Components

Given a point \((-2,2,-2)\) and a direction vector \(\langle 7, -6, 3 \rangle\). We need these components to form parametric equations.

Formulate Parametric Equations

The parametric equations for a line through a point \((x_0, y_0, z_0)\) with direction vector \(\langle a, b, c \rangle\) are:\[ x = x_0 + at \]\[ y = y_0 + bt \]\[ z = z_0 + ct \]Substituting the given point and vector:\[ x = -2 + 7t \]\[ y = 2 - 6t \]\[ z = -2 + 3t \]

Write Symmetric Equations

The symmetric equations of a line can be derived from the parametric equations:\[ \frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c} \]Substitute the respective values:\[ \frac{x + 2}{7} = \frac{y - 2}{-6} = \frac{z + 2}{3} \]

Key concepts

Symmetric Equations

Symmetric equations are another way to represent the equation of a line in space. These equations are derived from parametric equations, which describe a line by specifying a point on the line and a direction vector.
To transition from parametric to symmetric equations, each of the parametric equations is solved for the parameter, typically denoted as \( t \). For example:

• \( x = x_0 + at \) gives \( t = \frac{x - x_0}{a} \)
• \( y = y_0 + bt \) gives \( t = \frac{y - y_0}{b} \)
• \( z = z_0 + ct \) gives \( t = \frac{z - z_0}{c} \)

Combining these, we have the symmetric equation \( \frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c} \). This equation offers a holistic view of the line, showing how each coordinate changes relative to others. It can be particularly useful in 3D geometry, where visualizing all changes along the line simultaneously is beneficial.

Direction Vector

A direction vector is an essential component in representing lines in geometry, especially in three dimensions. This vector gives the orientation of the line, indicating which way the line extends from a given point.
Consider a line in space: each directional change in \( x, y, \) and \( z \) coordinates is determined by the corresponding components of the direction vector, typically written as \( \langle a, b, c \rangle \).

• The vector \( a \) tells us how the \( x \)-coordinate changes as we move along the line.
• The vector \( b \) affects how the \( y \)-coordinate changes.
• The vector \( c \) influences the \( z \)-coordinate.

The direction vector does not have to be a unit vector, meaning it does not need to be of length 1. It simply needs to provide the direction, and any scalar multiple of a direction vector is valid as it maintains the directionality of the line. This vector is fundamental in forming both parametric and symmetric equations of the line.

Parametric Representation

The parametric representation of a line offers a flexible way to describe a line via a set of equations involving a parameter, usually denoted as \( t \). This representation is especially valuable in 3D geometry, where it succinctly captures the trajectory of a line.
The parametric equations for a line are derived from a specific point \((x_0, y_0, z_0)\) and a direction vector \(\langle a, b, c \rangle\):

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

Here, \( t \) is a real number parameter that "travels" along the line, allowing you to calculate any point on the line by simply choosing a value for \( t \).
This representation is intuitive because it closely relates to the idea of moving along the line starting from a known point while following the direction specified by the vector. It offers clarity in situations where direct manipulation of coordinates is necessary, such as intersections and projections.

3D Geometry

Three-dimensional geometry, or 3D geometry, is the study of shapes and figures in a three-dimensional space defined by three axes: \( x \), \( y \), and \( z \). This realm of mathematics extends beyond the two-dimensional plane, incorporating depth and allowing for the exploration of more complex forms and alignments.
In 3D geometry, objects can move and be oriented toward any direction, making concepts like lines and planes more intricate. A line in 3D space not only has a slope but also a spatial direction, often defined by a point and a direction vector.
Key principles in 3D geometry include:

• Understanding line equations, both in parametric and symmetric forms, to navigate and compute locations in space.
• Analyzing planes, which are flat, two-dimensional surfaces extending indefinitely.
• Utilizing vectors to describe directions and magnitudes.

This field allows for the creation of more realistic models and simulations, as 3D geometry is frequently employed in computer graphics, engineering, and physics to represent and solve problems involving spatial relationships.