Problem 27 - Algebra and Trigonometry

Question

A description of a line is given. Find parametric equations for the line. The line crosses the \(z\) -axis where \(z=4\) and crosses the \(x y-\) plane where \(x=2\) and \(y=5 .\)

Step-by-Step Solution

Answer

The parametric equations are: \( x = 2t \), \( y = 5t \), \( z = 4 - 4t \).

Step 1: Identify Points

The line crosses the \( z \)-axis at the point \((0, 0, 4)\) and the \( xy \)-plane at \((2, 5, 0)\). These two points will be used to find the direction vector and parametric equations.

Step 2: Calculate Direction Vector

To find the direction vector of the line, calculate the difference between the two points. Direction Vector \( \mathbf{d} = (2 - 0, 5 - 0, 0 - 4) = (2, 5, -4) \).

Step 3: Write Parametric Equations

Using the point \((0, 0, 4)\) and the direction vector \((2, 5, -4)\) to write the parametric equations of the line. The parametric equations are: \( x = 0 + 2t \), \( y = 0 + 5t \), \( z = 4 - 4t \), where \( t \) is a parameter.

Key Concepts

Direction Vector

To understand parametric equations, it's essential to know about direction vectors. The direction vector gives us information about the orientation of a line in space. It describes both the direction and magnitude in which the line proceeds. In 3D geometry, this is represented as a vector, denoted by \( \mathbf{d} = (a, b, c) \). For our exercise, the direction vector is derived from the two points through which the line passes: one on the \( z \)-axis at \((0, 0, 4)\) and another on the \( xy \)-plane at \((2, 5, 0)\).
By calculating the difference in respective coordinates, \( \mathbf{d} = (2 - 0, 5 - 0, 0 - 4) \), we obtain \( (2, 5, -4) \). This vector describes the direction in which the line extends from the initial point. Understanding this vector allows us to construct the equation of a line in 3D space.

Line Equations

Line equations form the basis of describing straight lines in coordinate geometry. In parametric form, these equations display how every point on a line relates to a parameter, commonly represented by \( t \). This parameter helps in expressing each coordinate \( (x, y, z) \) as a function of \( t \). Given a point on a line and a direction vector, the parametric equations can be formulated as:
\[ x = x_0 + at \]
\[ y = y_0 + bt \]
\[ z = z_0 + ct \]
where \( (x_0, y_0, z_0) \) is a point on the line and \( (a, b, c) \) is the direction vector.
In the case of our exercise, using \((0, 0, 4)\) as the initial point and \((2, 5, -4)\) as the direction vector, the line's parametric equations are:
\( x = 0 + 2t \), \( y = 0 + 5t \), \( z = 4 - 4t \).
These equations describe every point on the line as \( t \) varies.

3D Coordinate Geometry

3D coordinate geometry extends the principles of 2D geometry into three-dimensional space, allowing us to model real-world objects more accurately. This field uses coordinates in the form \((x, y, z)\) to describe the position of points. Understanding 3D geometry involves not just plotting points, but also comprehending spatial relationships between lines, planes, and shapes.
When dealing with lines in 3D, parametric equations provide a powerful tool to express their paths. Each parameter value \( t \) corresponds to different points along the line. This is crucial in engineering, computer graphics, and physics, where precise modeling is needed.
In this example, the parametric equations \( x = 0 + 2t \), \( y = 0 + 5t \), \( z = 4 - 4t \) describe a line extending infinitely in both directions through 3D space, based on the relationship defined by the direction vector and the initial point. Mastering these concepts in 3D geometry enables you to solve complex spatial problems.