Question

Find the symmetric equations and the parametric equations of a line, and/or the equation of the plane satisfying the following given conditions. Line through (3,0,-5) and parallel to the line \(\mathbf{r}=(2,1,-5)+(0,-3,1) t\).

Short answer

Parametric: \[ x = 3, y = -3t, z = -5 + t \] Symmetric: \[ x = 3, \frac{y}{-3} = z + 5 \]

Step-by-step solution

Identify the Direction Vector

The line we are given is in the vector form \(\textbf{r} = (2,1,-5)+t(0,-3,1) \). The direction vector of this line is \((0, -3, 1)\). Since we need to find a line parallel to this one, our line will have the same direction vector \((0, -3, 1)\).

Use the Point to Write Parametric Equations

We know the line passes through the point \((3,0,-5)\) and has a direction vector \((0, -3, 1)\). So, the parametric equations of the line can be written as: \[ x = 3 + 0t \] \[ y = 0 - 3t \] \[ z = -5 + t \] \where t is a parameter.

Convert Parametric Equations to Symmetric Equations

To convert the parametric equations to symmetric form, eliminate the parameter t: Starting from the parametric equations: \[ x = 3 + 0t \] Since \x\ does not depend on \t\, the first equation is \x = 3\. Next: \[ y = 0 - 3t \ => t = -\frac{y}{3} \] And: \[ z = -5 + t \ => t = z + 5 \] Thus, the symmetric equations are: \[ x = 3 \] \[ \frac{y}{-3} = z + 5 \]

Key concepts

symmetric equations

Symmetric equations are a way to represent a line in 3D space without any parameters. They relate the coordinates (x, y, z) directly. While parametric equations use a parameter to define points on the line, symmetric equations eliminate it by combining the parametric equations into a single relation. To derive symmetric equations, start from the parametric form, eliminate the parameter, and rewrite the equations in a consistent form. For example, given parametric equations:

• \( x = 3 + 0t \)
• \( y = 0 - 3t \)
• \( z = -5 + t \)

we can eliminate the parameter, t, leading to the symmetric equations:

• \( x = 3 \)
• \( \frac{y}{-3} = z + 5 \)

.

parametric equations

Parametric equations describe a line in 3D space using a parameter, typically denoted as t, to express the coordinates x, y, and z independently. They are particularly useful because they provide a straightforward way to generate any point on the line. Given a point \(x_0, y_0, z_0\) and a direction vector \(a, b, c\), the parametric equations are usually written as:

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

For instance, for a line through (3, 0, -5) with direction vector (0, -3, 1), the parametric equations are:

• \( x = 3 + 0t \)
• \( y = 0 - 3t \)
• \( z = -5 + t \)

.

direction vector

The direction vector of a line indicates its orientation in 3D space. It is a vector that points in the direction of the line and has components that show how the line extends along the x, y, and z axes. From the equation of the line, the direction vector is usually the coefficient of the parameter t. In vector form, it often appears as part of an expression like \( \mathbf{r} = \mathbf{r_0} + t \mathbf{d} \)For example, if we're given the line equation \( \textbf{r} = (2,1,-5)+t(0,-3,1) \), the direction vector is \( (0, -3, 1) \). Directions vectors are crucial for both writing parametric equations and finding parallel lines.

3D geometry

3D geometry involves the study of shapes, lines, and points in three-dimensional space. Key concepts include vectors, lines, planes, and their equations. When solving problems in 3D geometry, you'll often use vectors to describe positions and directions. For instance:

• A line can be defined by a point and a direction vector.
• Planes can be represented by their normal vectors and a point on the plane.

Understanding how to represent and manipulate these elements is foundational in solving more complex geometrical problems. In our example, we use the initial point \(3, 0, -5\) and a direction vector \(0, -3, 1\) to form the equations for a line in 3D space.

vector algebra

Vector algebra deals with operations on vectors, such as addition, subtraction, and scalar multiplication, which are essential in 3D geometry. Vectors are defined by their magnitude and direction and are often represented in component form. Common vector operations include:

• Addition: \( \mathbf{A} + \mathbf{B} = (a_1 + b_1, a_2 + b_2, a_3 + b_3) \)
• Subtraction: \( \mathbf{A} - \mathbf{B} = (a_1 - b_1, a_2 - b_2, a_3 - b_3)\)
• Scalar Multiplication: \( k\mathbf{A} = (ka_1, ka_2, ka_3) \)

Vector algebra is used to find direction vectors and parametric equations by simplifying vector expressions and combining them appropriately.