Question

Find the shortest distance between the following pairs of parallel lines. $$ \begin{array}{l} \text { a. }\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{r} 2 \\ -1 \\ 3 \end{array}\right]+t\left[\begin{array}{r} 1 \\ -1 \\ 4 \end{array}\right] ; \\ {\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{l} 1 \\ 0 \\ 1 \end{array}\right]+t\left[\begin{array}{r} 1 \\ -1 \\ 4 \end{array}\right]} \\ \text { b. }\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{l} 3 \\ 0 \\ 2 \end{array}\right]+t\left[\begin{array}{l} 3 \\ 1 \\ 0 \end{array}\right] ; \\ \quad\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{r} -1 \\ 2 \\ 2 \end{array}\right]+t\left[\begin{array}{l} 3 \\ 1 \\ 0 \end{array}\right] \end{array} $$

Short answer

Shortest distances: a) \(\frac{3}{\sqrt{17}}\) b) 0, as both lines overlap perfectly.

Step-by-step solution

Identify the Direction Vector

Both parts of the exercise involve finding the shortest distance between pairs of parallel lines. For parallel lines, the direction vectors will be identical. In part (a), the direction vector for both lines is \([1, -1, 4]\). In part (b), the direction vector for both lines is \([3, 1, 0]\).

Determine a Vector Between the Lines

For part (a), find a point on each line: \([2, -1, 3]\) and \([1, 0, 1]\), respectively. Define a vector \(\mathbf{PQ}\) as \([1, 0, 1] - [2, -1, 3] = [-1, 1, -2]\). For part (b), take points \([3, 0, 2]\) and \([-1, 2, 2]\). Define a vector \(\mathbf{PQ}\) as \([-1, 2, 2] - [3, 0, 2] = [-4, 2, 0]\).

Project the Vector Onto the Normal Vector

To find the shortest distance, project the vector \(\mathbf{PQ}\) onto a normal vector to the line. The normal vector can be found as any vector perpendicular to the direction vector. In part (a), since the direction vector is \([1, -1, 4]\), a suitable normal vector is choice such as \([-4, 0, 1]\). For part (b), since the direction vector is \([3, 1, 0]\), a suitable normal vector is \([0, 0, 1]\).

Calculate the Length of the Projection

For part (a): Calculate the dot product between \(\mathbf{PQ} = [-1, 1, -2]\) and the normal vector \([-4, 0, 1]\). The projection magnitude is \(\frac{\mathbf{PQ} \cdot \mathbf{N}}{\|\mathbf{N}\|}\). For part (b): Calculate the dot product between \(\mathbf{PQ} = [-4, 2, 0]\) and \([0, 0, 1]\). Again, treat as a projection to find distance.

Simplify the Distance Calculations

For part (a), perform \(\frac{[-1, 1, -2] \cdot [-4, 0, 1]}{\sqrt{16+1}}\) to get a distance of \(\frac{3}{\sqrt{17}}\). For part (b), perform the same operation and deduce that the shortest distance is \(|-4 \cdot 0 + 2 \cdot 0 + 0 \cdot 1| = 0\) since \([0, 0, 1]\) is perpendicular to \([-4, 2, 0]\), thus already minimized.

Key concepts

Direction Vector

The direction vector is a key concept when analyzing parallel lines in 3D space. It gives us information on how a line extends in space. This vector remains constant for parallel lines, indicating that they run indefinitely alongside each other without intersecting.

In the provided problem, each set of parallel lines shares the same direction vector. For example, the direction vector for the lines in part (a) is \([1, -1, 4]\). Meanwhile, lines in part (b) share the direction vector \([3, 1, 0]\). It is crucial to identify these vectors because they help determine which vectors are parallel and subsequently guide us in calculating the shortest distance between the two lines.

The direction vector essentially describes the trajectory of the line, revealing its slope or angle in a three-dimensional space. Understanding this will form the foundation upon which we figure out distances later on.

Normal Vector

A normal vector is perpendicular to a given plane or line. This right angle relationship makes the normal vector an instrumental concept in calculating distances, especially the shortest distance to a line.

In problems involving lines, such as finding the shortest distance between parallel lines, we derive a normal vector from the direction vector by ensuring it has a perpendicular orientation to the direction of the lines.

For instance, in part (a) with a direction vector of \([1, -1, 4]\), a corresponding normal vector could be \([-4, 0, 1]\). For part (b), where the direction vector is \([3, 1, 0]\), possible normal vector choices include \([0, 0, 1]\). These vectors serve as tools, enabling projection operations that directly assist in finding the shortest possible distance between the lines.

Dot Product

The dot product is a mathematical operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. It combines algebra and geometry, scaling one vector along another.

In the context of our exercise, the dot product is used to measure how much of one vector goes in the direction of another. This measurement is crucial for calculating the projection — and subsequently, the shortest distance. The dot product of two vectors \(\mathbf{A}=[a_1, a_2, a_3]\) and \(\mathbf{B}=[b_1, b_2, b_3]\) is computed as \(a_1b_1 + a_2b_2 + a_3b_3\).

In part (a), the dot product of \([-1, 1, -2]\) and \([-4, 0, 1]\) helps measure the extent of \([-1, 1, -2]\) in the direction of the normal vector. This operation is integral to finding the distance between parallel lines, as it connects the vectors geometrically.

Projection of Vectors

Projection of one vector onto another is an operation that casts one vector onto the "shadow" of another vector, as if it were shining a light perpendicularly. This mathematical concept allows us to identify components of vectors that align with each other.

In relation to the exercise, projecting the vector connecting two points from each line onto the normal vector helps find the shortest distance between the parallel lines. First, compute the dot product of the vector pointing between two lines \(\mathbf{PQ}\) and the normal vector \(\mathbf{N}\). Then divide this by the magnitude of the normal vector, calculated as \(\sqrt{n_1^2+n_2^2+n_3^2}\), giving you the projection's length.

For example, in part (a), the projection of \([-1, 1, -2]\) onto \([-4, 0, 1]\) gives insight into how these vectors relate spatially, ultimately revealing the minimum distance between the parallel lines.

• Projection helps turn complex 3D operations into more manageable, simpler calculations.
• It aids in visualizing spatial interactions and distances.