Question

Find the symmetric equations (5.6) or \((5.7)\) and the parametric equations \((5.8)\) of a line, and/or the equation (5.10) of the plane, satisfying the following given conditions. Line through \((0,-2,4)\) and \((3,-2,-1)\)

Short answer

Parametric: \(x = 0 + 3t\), \(y = -2 + 0t\), \(z = 4 - 5t\); Symmetric: \(\frac{x}{3} = \frac{y + 2}{0} = \frac{z - 4}{-5}\).

Step-by-step solution

- Determine Direction Vector

Calculate the direction vector \(\textbf{d}\) of the line by subtracting the coordinates of the initial point \(P_0 = (0, -2, 4)\) from the coordinates of another point \(P_1 = (3, -2, -1)\). This leads to: \(\textbf{d} = P_1 - P_0 = (3 - 0, -2 - (-2), -1 - 4) = (3, 0, -5)\).

- Write Parametric Equations

The parametric equations of the line can be expressed using the point \(P_0 = (0, -2, 4)\) and the direction vector \(\textbf{d} = (3, 0, -5)\). This leads to: \(x = 0 + 3t\), \(y = -2 + 0t\), \(z = 4 - 5t\), where \(t\) is a parameter.

- Write Symmetric Equations

Convert the parametric equations into symmetric form. Use the relations: \(\frac{x - x_0}{d_x},\ \frac{y - y_0}{d_y},\ \frac{z - z_0}{d_z}\). Here \((x_0, y_0, z_0) = (0, -2, 4)\) and \(d = (3, 0, -5)\). Thus, \(\frac{x - 0}{3} = \frac{y + 2}{0} = \frac{z - 4}{-5}\).

Key concepts

Parametric Equations

Parametric equations are a way to express a line using a set of equations based on a parameter, usually denoted as \(t\). In the context of a line, we utilize a point the line passes through, and a direction vector. By adding the direction vector, scaled by the parameter \(t\), to the coordinates of the base point, we describe the entire line.

For instance, consider a line through the point \((0, -2, 4)\), with a direction vector of \((3, 0, -5)\). The parametric equations of the line will be:

• \(x = 0 + 3t\)
• \(y = -2 + 0t\) (which simplifies to \(y = -2\))
• \(z = 4 - 5t\)

Here, \(t\) can be any real number, as it parameterizes the entire line.

Symmetric Equations

Symmetric equations provide another way to represent a line, linking the coordinates directly through ratios. They derive from parametric equations and are particularly useful in removing the parameter \(t\).

• First, we take our parametric forms: \(x = 0 + 3t\), \(y = -2 + 0t\), \(z = 4 - 5t\).
• We then express \(t\) from each of these equations:
• \(t = \frac{x}{3}\)
• \(t = \frac{y + 2}{0}\) (undefined, thus we often skip the zero direction)
• \(t = \frac{z - 4}{-5}\)

This results in the symmetric equations:\( \frac{x}{3} = \frac{z - 4}{-5} \).

Direction Vector

A direction vector determines the direction of a line in space. It is found by subtracting the coordinates of the starting point from another point on the line.

Consider the line through points \((0, -2, 4)\) and \((3, -2, -1)\). The direction vector \( \textbf{d} \) is computed as:
\[ \textbf{d} = (3 - 0, -2 - (-2), -1 - 4) = (3, 0, -5) \]
This vector \( (3, 0, -5) \) represents the direction in which the line extends in 3D space.

Linear Algebra

Linear algebra often deals with equations describing geometric objects like lines and planes. It helps in converting different forms of equations and manipulating vectors.

For instance:

• Equations of a line can be converted between parametric and symmetric forms using vector operations.
• Cross and dot products of vectors determine angles and orthogonality between lines and planes.
• Transformations in linear algebra rotate, translate, or scale geometric objects.

In essence, linear algebra provides a toolkit for handling equations that describe lines and planes in multidimensional space.

Plane Equation

The equation of a plane in 3D is often given by \(ax + by + cz = d\), where \(a, b, c\) are the coefficients derived from the normal vector to the plane, and \(d\) is a constant.

To find a plane equation, we need:

• A point on the plane (to help determine \(d\)).
• A normal vector to the plane.

If a plane passes through point \((x_1, y_1, z_1)\) with a normal vector \((a, b, c)\), the plane equation is:
\[ a(x - x_1) + b(y - y_1) + c(z - z_1) = 0 \]

This general form helps in easily determining whether a given point lies on the plane or not.