Question

Find sets of (a) parametric equations and (b) symmetric equations of the line through the point parallel to the given vector or line. (For each line, write the direction numbers as integers.) $$ (-2,0,3) \quad \mathbf{v}=2 \mathbf{i}+4 \mathbf{j}-2 \mathbf{k} $$

Short answer

The parametric equations are \(x = -2 + 2t\), \(y = 4t\), \(z = 3 - 2t\), and the symmetric equations are \(\frac{x + 2}{2} = \frac{y}{4} = \frac{z-3}{-2}\).

Step-by-step solution

Identify Values

First, it's necessary to identify the components of the given direction vector \(\mathbf{v}\) as well as the coordinates of the given point. Here, the vector \(\mathbf{v}\) has components of 2, 4, and -2, represented by \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) respectively. And the given point is (-2,0,3).

Write Parametric Equations

The parametric equations for a line in three dimensions can be written as \(x = x_0 + at\), \(y = y_0 + bt\), \(z = z_0 + ct\), where \((a,b,c)\) are the direction numbers or components of the direction vector, \((x_0, y_0, z_0)\) is a point on the line, and \(t\) is the parameter. Substitute the identified values into these equations, we have: \(x = -2 + 2t\), \(y = 0 + 4t\), and \(z = 3 - 2t\).

Write Symmetric Equations

The symmetric equations can be written as \(\frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c}\). Substitute the identified values into these equations, we have: \(\frac{x + 2}{2} = \frac{y}{4} = \frac{z-3}{-2}\).

Key concepts

Parametric Equations in Calculus

Parametric equations provide a convenient way to represent curves and lines in calculus, especially when dealing with three dimensions. Unlike traditional Cartesian equations that express y in terms of x, parametric equations describe a path by using a third variable, called a parameter, usually denoted as 't'.

In the context of a line, parametric equations define the x, y, and z coordinates separately in terms of this parameter. For a line, these equations often have the form:

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

where \((a, b, c)\) are the components of the direction vector, \((x_0, y_0, z_0)\) is a specific point the line passes through, usually referred to as the position vector, and 't' can be any real number. As 't' varies, \((x, y, z)\) trace the path of the line in space. This is a powerful representation, as it easily extends to higher dimensions and captures the essence of motion or growth along a path.

Direction Vector

A direction vector is fundamental in representing lines in a vector space. It reveals the orientation of the line and dictates its slope, directly influencing the trajectory the line takes through space. Specifically, the direction vector consists of components that define how much the line moves along the x, y, and z axes for a unit increase in the parameter 't'.

In our exercise, the given direction vector is \(2 \mathbf{i}+4 \mathbf{j}-2 \mathbf{k}\). The coefficients of \(\mathbf{i}, \mathbf{j}, \mathbf{k}\) (namely 2, 4, and -2) represent the direction numbers 'a', 'b', and 'c', respectively. These numerals give us the necessary information to determine the line's direction. By incorporating these elements into the parametric form of the equations, we gain a set of equations that entirely and uniquely determine the path of the line.

Symmetric Equations of a Line

Symmetric equations serve as an alternate way of depicting a line in a three-dimensional space when the direction ratios or direction cosines are known. This form is particularly useful for identifying points where the line may intersect with planes or other geometric entities.

The general form of symmetric equations for a line is:\[\frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c}\]. Here, they succinctly capture the essence of the line's behavior and provide a clear visual of how the line extends through space relative to a fixed point. To construct these equations, we rely on the same initial variables as needed for parametric equations: the direction vector's components and a point the line passes through, effectively translating the line's dynamics into a proportionality between the coordinates and their respective directional movements.

Three-Dimensional Coordinate Systems

Three-dimensional coordinate systems expand upon the familiar two-dimensional Cartesian plane, adding a z-axis which enables a representation of space that's closer to our real-world experience. In this system, coordinates are given as ordered triples \((x, y, z)\), defining a point's position in terms of its distances from three perpendicular axes.

For students visualizing lines and curves in this space, it's crucial to grasp not just the static positions but also how entities like lines move or change direction as one travels along them. Parametric and symmetric equations both aid in this understanding, providing dynamic descriptions that, when plotted, reveal the spatial trajectory of lines. With these tools, concepts like intersections, angles between lines, and distances can be more deeply comprehended within the context of three-dimensional geometry.