Problem 72 - Thomas Calculus

Question

Suppose \(L_{1}\) and \(L_{2}\) are disjoint (nonintersecting) nonparallel lines. Is it possible for a nonzero vector to be perpendicular to both \(L_{1}\) and \(L_{2} ?\) Give reasons for your answer.

Step-by-Step Solution

Answer

No, a nonzero vector cannot be perpendicular to disjoint, nonparallel lines as they lie in different planes.

Step 1: Understanding Line Conditions

Lines \(L_1\) and \(L_2\) are described as disjoint and nonparallel. Disjoint means they do not intersect anywhere in space, and nonparallel indicates they do not run in the same direction. Such conditions imply that the two lines lie in different planes.

Step 2: Exploring Perpendicularity to Lines

A vector is perpendicular to a line if it forms a 90-degree angle with every vector that lies on that line. For a vector to be perpendicular to two lines simultaneously, within the same plane, the lines must intersect or be parallel. However, \(L_1\) and \(L_2\) are disjoint and nonparallel, which means they are not in the same plane.

Step 3: Analyzing the Spatial Arrangement

Since \(L_1\) and \(L_2\) are in different planes, this geometrically implies that no single vector can lie orthogonal to both lines at the same time, because the spatial orientation of each line is such that their normal vectors do not coincide.

Step 4: Concluding the Possibility

Given the lines' spatial arrangement, it is impossible for a nonzero vector to be perpendicular to both \(L_1\) and \(L_2\). This is due to the lack of a common plane, since two sets of vectors from different planes cannot be simultaneously orthogonal to a single vector.

Key Concepts

Disjoint Lines

Disjoint lines are lines that do not intersect each other anywhere in the three-dimensional space. This has an important implication for their spatial relationship because it means that the lines lie in entirely different planes. They never share common points.
Disjoint lines are often confused with parallel lines, but they are intrinsically different. Parallel lines, if extended infinitely, will never meet, yet they reside within the same plane. In contrast, disjoint lines live in separate planes altogether.

• Disjoint lines do not intersect.
• They do not lie on the same plane.

This distinct spatial relationship affects how vectors can interact with these lines, particularly in terms of perpendicularity, as we'll explore in later sections.

Nonparallel Lines

When lines are described as nonparallel, they are not aligned in the same direction. This characteristic implies that if extended indefinitely, they do not remain equidistant from one another as parallel lines do. Nonparallel lines might intersect each other, but in the case of disjoint nonparallel lines, we've established they reside in different planes.
Nonparallelism is a key factor when considering the geometric properties of lines in space. It clarifies that the lines have distinct directional vectors, and this separates them spatially.

• Nonparallel lines can intersect but do not, if disjoint.
• They indicate differing directional vectors.

This affects the feasibility of finding a common vector that is perpendicular to both lines.

Vector Orthogonality

Vector orthogonality is a concept where two vectors are perpendicular to each other, forming a right-angle intersection. A vector is orthogonal to a line if it creates a 90-degree angle with every vector that makes up the line.
For two lines to share a common vector that is orthogonal, they would usually need to be either parallel or intersecting, which allows a single plane of multiple vectors to intersect at right angles with this common vector. In cases where lines are disjoint and in separate planes, it isn't possible for one vector to maintain orthogonality with vectors in both lines simultaneously.

• An orthogonal vector forms a 90-degree angle with another vector.
• Orthogonality requires a common plane between the vectors.

This property is crucial when analyzing possible vectors that could theoretically relate to both lines.

Spatial Geometry

Spatial geometry deals with the properties and dimensions of objects in three-dimensional space. This branch of mathematics explores how different shapes, lines, and surfaces relate to each other.
Understanding spatial geometry is essential when considering disjoint and nonparallel lines because it provides a framework for their positioning in space. When lines exist in different planes, they define unique spatial regions with no overlap—making it impossible for a single vector to conform to certain geometric constraints like orthogonality to both lines.

• Spatial geometry involves 3D space considerations.
• It studies dimensions and relationships among lines and planes.

Given these spatial considerations, geometry informs us why finding a common perpendicular vector to two disjoint, nonparallel lines is infeasible.