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In the study of vectors, a direction vector is crucial for understanding how a line extends in three-dimensional space. A line can be described by its path through two points. The direction vector acts like a pointer, indicating the path's trajectory from one point to another. For any given two points, say \(\mathbf{r}_1\) and \(\mathbf{r}_2\), the direction vector (\(\mathbf{d}\)) is calculated by simply subtracting one position vector from the other: \[ \mathbf{d} = \mathbf{r}_2 - \mathbf{r}_1. \] With coordinates, this vector becomes tangible as it gives the components along each axis. In our example, we found the direction vector to be \(0\hat{\mathbf{i}} - 4 \hat{\mathbf{j}} + 3 \hat{\mathbf{k}}\). It essentially tells us how much the line moves parallel to each coordinate axis when moving from the first point to the second.

The parametric equation of a line is a versatile tool that represents a line's path in vectors by considering a point on the line and extending it in the direction of a direction vector. The idea is simple: if you know where you begin (a fixed point \(\mathbf{r}_1\)) and the direction you're heading (the direction vector \(\mathbf{d}\)), you can describe any point on the line with a scalable parameter \(t\), which stretches or shrinks the direction vector accordingly. The parametric form is given by: \[ \mathbf{r} = \mathbf{r}_1 + t \mathbf{d}. \] This equation effectively maps a line as a set of points that can be generated by varying \(t\). In our application, the initial point \(\hat{\mathbf{i}} + \hat{\mathbf{j}} - 2\hat{\mathbf{k}}\) combined with the direction vector yields a line that can be explored by changing the value of \(t\).

A position vector is a critical concept that signifies the location of a point in space in relation to the origin. Think of it as an arrow that stretches from the origin to a specific point. It is typically denoted as \(\mathbf{r}\), and written in terms of the unit vectors \(\hat{\mathbf{i}}, \hat{\mathbf{j}},\) and \(\hat{\mathbf{k}}\) in three-dimensional space. Position vectors give complete and precise information on location in spatial terms, making them foundational in understanding vector spaces. In practice, a position vector like \(\mathbf{r} = \hat{\mathbf{i}} + \hat{\mathbf{j}} - 2\hat{\mathbf{k}}\) indicates moving one unit along the x-axis, one unit along the y-axis, and two units backward along the z-axis from the origin. This understanding is key when setting up parametric equations or determining how two points relate to each other on a line.

The distance formula allows us to figure out how far apart two points are in space using their coordinates. For two points with position vectors, the formula is used to calculate the magnitude of the difference between these vectors. When spaced out mathematically, finding the distance involves computing the square root of the sum of the squared differences of their individual components. In practice, if two position vectors \(\mathbf{r}_1\) and \(\mathbf{r}_3(t)\) are involved, we can use the equation: \[\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}.\] For our case, the distance helps determine how far new points on a line are from original locations by solving for \(t\), ensuring the distance equals our desired measurement, such as a unit distance. This operation forms the basis for solving many practical problems in geometry and vector algebra.