Question

Find the distance between the two lines. $$ \begin{array}{l} \vec{\ell}_{1}(t)=\langle 0,0,1\rangle+t\langle 1,0,0\rangle \\ \vec{\ell}_{2}(t)=\langle 0,0,3\rangle+t\langle 0,1,0\rangle \end{array} $$

Short answer

The distance between the lines is 2 units.

Step-by-step solution

Understand the Problem

The problem requires us to find the distance between two skew lines in 3D space, defined by their parametric equations. Since these lines are neither parallel nor intersecting, they are skew and do not lie on the same plane.

Identify Direction Vectors and Points

For each line provided, identify a point and a direction vector.\For \( \vec{\ell}_1(t) \) the point is \( P_1(0,0,1) \) and the direction vector is \( \langle 1,0,0 \rangle \).\For \( \vec{\ell}_2(t) \) the point is \( P_2(0,0,3) \) and the direction vector is \( \langle 0,1,0 \rangle \).

Find the Cross Product of Direction Vectors

Find the cross product of the direction vectors of the two lines: \( \langle 1,0,0 \rangle \times \langle 0,1,0 \rangle \). This gives the vector \( \langle 0,0,-1 \rangle \).

Calculate the Magnitude of the Cross Product

Calculate the magnitude of the cross product. For the vector \( \langle 0,0,-1 \rangle \), the magnitude is \( |\langle 0,0,-1 \rangle| = 1 \).

Find the Vector Between Points on Each Line

Calculate the vector between the points \( P_1(0,0,1) \) and \( P_2(0,0,3) \): \( \langle 0-0, 0-0, 1-3 \rangle = \langle 0, 0, -2 \rangle \).

Project the Vector Onto Cross Product

Find the projection of \( \langle 0,0,-2 \rangle \) onto \( \langle 0,0,-1 \rangle \). It is calculated as \( \text{proj}_{\langle 0,0,-1 \rangle}(\langle 0,0,-2 \rangle) = \frac{\langle 0,0,-2 \rangle \cdot \langle 0,0,-1 \rangle}{1} = 2 \).

Conclusion

The projection value gives the shortest distance between the two lines, which is 2.

Key concepts

Skew Lines

In geometry, skew lines are pairs of lines that do not share a common plane. Unlike parallel lines, which remain equidistant and lie in the same plane, skew lines go in different directions and are found in three-dimensional space. They do not intersect and are not parallel, which makes them interesting for understanding spatial relationships in 3D geometry. To identify skew lines, we must look at their directions and ensure they neither cross each other nor maintain a set distance in any plane. Skew lines only exist in 3D environments, showcasing their importance in real-world scenarios such as architecture or physics where thinking outside the plane is necessary.

Cross Product

The cross product is a mathematical operation that takes two vectors in 3D space and returns another vector that is perpendicular to the plane formed by the original vectors. Given two vectors, \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \), their cross product \( \mathbf{a} \times \mathbf{b} \) is calculated as:

• \( \mathbf{a} \times \mathbf{b} = \langle a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1 \rangle \)

The resulting vector is crucial for finding distances or areas because it is normal to the plane, and this perpendicularity can simplify many problems. In the context of skew lines, the cross product helps in determining the shortest distance between them. The magnitude of this resultant vector also gives the area of the parallelogram formed by the original vectors.

Parametric Equations

Parametric equations are a way to represent geometric figures, like lines and curves, using variables—often called parameters. In the context of 3D space, a line can be expressed using a point and a direction vector, with a parameter \( t \) that varies:

• For example, the parametric equation of a line might look like \( \mathbf{r}(t) = \mathbf{p} + t\mathbf{d} \), where \( \mathbf{p} \) is a point on the line, \( \mathbf{d} \) is the direction vector, and \( t \) is the parameter.

This representation is powerful because it offers flexibility in navigating any point on the line by simply adjusting \( t \). By knowing both the starting point and the direction, any position along the line can be computed. This becomes particularly useful in defining lines in 3D space where traditional Cartesian equations would become unwieldy.

3D Space

Three-dimensional (3D) space represents the environment we naturally live in, where objects have length, width, and height. Unlike 2D space, which is confined to flat surfaces, 3D space involves movement in a third dimension. Understanding 3D space is critical in many fields such as physics, engineering, and computer graphics, as it allows for more precise modeling of the real world.

• Key elements in 3D space include points, lines, planes, and vectors. These elements help describe the positions and movements of objects.
• 3D space offers more complex geometric configurations, such as skew lines, which cannot be described within a 2D plane.

In problems involving distances, particularly between lines, the additional dimension requires a deeper understanding of spatial relationships and vector operations, which are fundamental in calculating distances or angles.