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In three-dimensional space, a direction vector is fundamental for defining the orientation of a line. It essentially acts like a compass, indicating the direction in which the line is pointing.
The direction vector is usually represented by the components enclosed in angle brackets or as a combination of unit vectors such as \(\mathbf{i}, \mathbf{j}, \) and \(\mathbf{k}\).
For instance, in the given problem, the direction vector is \(3\mathbf{i} - 4\mathbf{j} + 7\mathbf{k}.\) This vector shows us how the line moves in the x, y, and z directions.

• The first component, 3, means the line increases by 3 units along the x-axis for each step.
• Similarly, -4 indicates a decrease along the y-axis, while 7 shows an increase along the z-axis.

This clarity is vital when calculating the line's trajectory in space, as it distinctly specifies the direction regardless of the line’s starting point.

The point-vector form of a line in three-dimensional space is an elegant way to represent the line using both its direction and a single known point.
This form is expressed mathematically as \(\textbf{r} = \textbf{a} + \lambda \textbf{d},\) where \(\textbf{r}\) is the position vector of any point on the line, \(\textbf{a}\) is a known point on the line, and \(\textbf{d}\) is the direction vector. The parameter \(\lambda\) is a scalar that allows one to move along the line.

• In our current example, \(\textbf{a}\) is the point \(-2\mathbf{i} + 1\mathbf{j} + 4\mathbf{k}\).
• This form combines both the direction of the line and its position in space.

The point-vector form makes it easy to visualize and compute various positions on the line. By choosing different values for \(\lambda\), you can find any other point along this infinite line, starting from the initial point and moving in the direction given by the direction vector.

Parametric equations are a set of equations that express the coordinates of the points on a line—or any curve—as a function of one or more parameters.
These provide a flexible form of representing lines, particularly in three-dimensional space, by indicating how x, y, and z coordinates change based on a parameter \(\lambda\).
Given the vector equation \(\textbf{r} = \textbf{a} + \lambda \textbf{d},\), it can be split into separate equations for each coordinate.
For our problem, substituting the given point and direction vector, we obtain the parametric equations:

• \(x = -2 + 3\lambda\),
• \(y = 1 - 4\lambda\),
• \(z = 4 + 7\lambda\).

These equations describe how adding scalar multiples of \(\lambda\) to each coordinate helps navigate along the line. As \(\lambda\) changes, the corresponding point on the line shifts, allowing you to find any point on the line. This systematic approach is widely used in geometry and physics to analyze line properties or solve problems involving lines in space.