Problem 76 - University Calculus: Early Transcendentals

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 nonparallel lines.

Step 1: Understanding the Problem

We have two nonparallel, disjoint lines (they neither intersect nor overlap in direction), and we need to check if a nonzero vector can be perpendicular to both lines. A vector is perpendicular to a line if it forms a 90-degree angle with it.

Step 2: Defining a Vector Perpendicular to a Line

To have a vector perpendicular to a line, it must be orthogonal to the directional vector of that line. Let's denote the directional vector of \( L_1 \) as \( \mathbf{a} \) and for \( L_2 \) as \( \mathbf{b} \). A vector \( \mathbf{v} \) is perpendicular to \( L_1 \) if \( \mathbf{v} \cdot \mathbf{a} = 0 \) and to \( L_2 \) if \( \mathbf{v} \cdot \mathbf{b} = 0 \).

Step 3: Exploring the Perpendicular Condition

The conditions \( \mathbf{v} \cdot \mathbf{a} = 0 \) and \( \mathbf{v} \cdot \mathbf{b} = 0 \) imply that \( \mathbf{v} \) should be orthogonal to both \( \mathbf{a} \) and \( \mathbf{b} \). However, \( \mathbf{a} \) and \( \mathbf{b} \) are directions for nonparallel lines, indicating that they are linearly independent.

Step 4: Solution by Linear Independence

Because \( \mathbf{a} \) and \( \mathbf{b} \) are linearly independent, the only vector \( \mathbf{v} \) that satisfies both perpendicular conditions \( \mathbf{v} \cdot \mathbf{a} = 0 \) and \( \mathbf{v} \cdot \mathbf{b} = 0 \) is the zero vector. But since zero is not a nonzero vector, it is not possible for a nonzero vector to be perpendicular to both \( L_1 \) and \( L_2 \).

Key Concepts

Linear Independence

In mathematics, two vectors are considered linearly independent if no linear combination of these vectors equals zero unless all coefficients are zero.
Simply put, you cannot express one vector as a multiple of the other.

• If two vectors are nonparallel, like our lines' directional vectors \( \mathbf{a} \) and \( \mathbf{b} \), they are linearly independent.
• Therefore, they span a plane in space.
• This plane is where every linear combination of \( \mathbf{a} \) and \( \mathbf{b} \) lives.

If you imagine stretching and rotating these directional vectors through all possible angles, they would mark out a flat surface—or plane—in space.
Because of their independence, finding another vector that is simultaneously perpendicular to both is impossible, except for the zero vector.

Orthogonality

Orthogonality refers to vectors being at right angles to each other.
In mathematical terms, two vectors \( \mathbf{u} \) and \( \mathbf{v} \) are orthogonal if their dot product equals zero: \( \mathbf{u} \cdot \mathbf{v} = 0 \).

• For a vector to be perpendicular to a line, it must be orthogonal to the line's directional vector.
• In our exercise, a vector \( \mathbf{v} \) is required to be orthogonal to both \( \mathbf{a} \) and \( \mathbf{b} \).

Both conditions \( \mathbf{v} \cdot \mathbf{a} = 0 \) and \( \mathbf{v} \cdot \mathbf{b} = 0 \) must be satisfied for orthogonality to occur.
While these conditions restrict \( \mathbf{v} \) to lie in a plane orthogonal to each directional vector independently, the linear independence of \( \mathbf{a} \) and \( \mathbf{b} \) means their only shared orthogonal vector is the zero vector. This makes it impossible to have a nonzero vector that fulfills both conditions.

Directional Vectors

Directional vectors define the direction of lines or planes in space.
In a three-dimensional space, a directional vector is like an arrow pointing out where a line goes.

• The directional vector of a line is used to calculate perpendicular vectors, those that meet the line at a right angle.
• In our scenario, \( \mathbf{a} \) and \( \mathbf{b} \) are directional vectors of nonparallel lines \( L_1 \) and \( L_2 \).
• Since the lines are nonparallel, their directions do not overlap.

When solving problems involving perpendicular vectors, the directional vectors play a key role.Whether you're looking at geometry or vector calculus, understanding how to use directional vectors is important.
They help us determine relationships like perpendicularity, crucial in proving the limitations set by linear independence and orthogonality.