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 both disjoint, nonparallel lines.

Step 1: Understanding Perpendicular Vectors

A vector is perpendicular to a line if it forms a 90-degree angle with the line. For a vector to be perpendicular to two lines simultaneously, there must be a point where such a vector can touch both lines.

Step 2: Nature of Disjoint Nonparallel Lines

Disjoint nonparallel lines do not intersect at any point. Therefore, there is no single point where a vector can simultaneously be perpendicular to both lines.

Step 3: Perpendicularity Condition

For a vector to be perpendicular to both lines, there would need to be a common plane or point involving both lines that could support such a vector. With disjoint lines being nonparallel, no such condition exists.

Step 4: Conclusion of Vector Existence

It is impossible for a single nonzero vector to be perpendicular to both disjoint, nonparallel lines due to the lack of a common intersection or plane.

Key Concepts

Disjoint Nonparallel Lines

Disjoint nonparallel lines are a fascinating geometric concept. In geometry, lines can either be parallel, intersecting, or disjoint. Disjoint lines do not intersect at any point. This means that they run in different planes or directions without meeting. When they are identified as nonparallel, it emphasizes that they are not moving in the same direction or maintaining an equidistant space from each other.

In a three-dimensional space, this can be visualized almost like two roads on an overpass that neither meet nor run parallel. Disjoint nonparallel lines are therefore quite different from intersecting lines, which cross each other at some point.

When understanding vectors in relation to such lines, it is crucial to remember that since they are disjoint and nonparallel, we cannot find a single point or a plane where interactions happen, including perpendicular vectors touching both.

Perpendicularity Condition

The perpendicularity condition involves a vector forming a right angle (90 degrees) with a line or plane. This is quite common in geometry, showing orthogonality. A vector being perpendicular to a line means if you were to "drop" the vector towards the line, it would touch it creating a perfect "T" shape without any tilt.

Imagine crossing the letter "T"; the vertical line is perpendicular to the horizontal line, demonstrating clear perpendicularity. For two lines, a single vector can be perpendicular to both only if there's a common plane or single point where such a situation is geometrically possible.

However, when it comes to disjoint and nonparallel lines, this condition becomes impossible to satisfy. Since these lines do not share a point or plane, the prospect of having a vector perpendicular to both at the same time is geometrically unfeasible.

Vector Analysis

In vector analysis, we study how vectors interact with different geometric entities like lines and planes. A vector is a mathematical entity possessing both direction and magnitude, often represented as an arrow. It can define a line by determining its direction and length.

For a vector to be perpendicular to a line, its dot product with the directional vector of the line must be zero. This is fundamental in understanding perpendicularity in vector analysis.

• Dot Product: Determines angles between vectors.
• Perpendicularity Achieved: When the dot product equals zero.

When working with lines that are disjoint and nonparallel, this analysis shows that no single vector can simultaneously have zero dot products with the directional vectors of the two given lines. As the lines are disjoint, they lack a shared plane or intersection, thus a vector cannot be perpendicular to both.