Question

Let the vectors \(\overrightarrow{P Q}, \overrightarrow{Q R}, \overrightarrow{R S}, \overrightarrow{S T}, \overrightarrow{T U}\) and \(\overrightarrow{U P}\) represent the sides of a regular hexagon. STATEMENT- \(1: \overrightarrow{P Q} \times(\overrightarrow{R S}+S \vec{T}) \neq \overrightarrow{0}\) because STATEMENT- \(2: \overrightarrow{P Q} \times \overrightarrow{R S}=\overrightarrow{0}\) and \(\overrightarrow{P Q} \times \overrightarrow{S T} \neq \overrightarrow{0}\) (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1 (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True

Short answer

Statement-1 is true and Statement-2 is true, but Statement-2 is not a correct explanation for Statement-1, which makes option (B) the correct answer.

Step-by-step solution

Understanding the Geometry of Regular Hexagon

In a regular hexagon, the opposite sides are parallel and equal in length. Therefore, vector \(\overrightarrow{PQ}\) is parallel to vector \(\overrightarrow{RS}\), and vector \(\overrightarrow{ST}\) is parallel to vector \(\overrightarrow{UP}\).

Analyze Statement-2

Since \(\overrightarrow{PQ}\) is parallel to \(\overrightarrow{RS}\), their cross product is zero: \(\overrightarrow{PQ} \times \overrightarrow{RS} = \overrightarrow{0}\). Additionally, since \(\overrightarrow{ST}\) is parallel to \(\overrightarrow{UP}\), and \(\overrightarrow{PQ}\) is not parallel to \(\overrightarrow{ST}\), it follows that \(\overrightarrow{PQ} \times \overrightarrow{ST} eq \overrightarrow{0}\).

Evaluate Statement-1

Considering \(\overrightarrow{PQ}\) is not parallel to \(\overrightarrow{ST}\), the cross product of \(\overrightarrow{PQ}\) and the sum of vectors \(\overrightarrow{RS} + \overrightarrow{ST}\) is not zero because one of the terms of the cross product, \(\overrightarrow{PQ} \times \overrightarrow{ST}\), is nonzero.

Conclude the Correct Option

Since both statements are true but Statement-2 does not provide a correct explanation for Statement-1, the correct answer is option (B): Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.

Key concepts

Regular Hexagon Properties

A regular hexagon is a six-sided polygon that exhibits remarkable symmetry and geometric properties that are crucial in various areas of mathematics and applications. One of the key characteristics is that all its sides have equal length and all its interior angles are equal, each measuring 120 degrees. Another important property is that each side is parallel to the side opposite to it, which means the vectors representing these sides are also parallel. In a vector representation of a regular hexagon, opposite sides can be described as equivalent vectors with the same magnitude but potentially different directions depending on their orientation in space.

This property of parallel sides is essential when discussing vector operations such as the vector cross product, which will always lead to zero when performed on parallel vectors, highlighting one of the foundations of vector algebra relating to parallelism.

In the context of the given exercise, the parallelism of opposite sides of the hexagon directly influences the outcomes of cross product operations and forms an integral part of understanding the vector algebra involved in the problem.

Vector Parallelism

Vector parallelism signifies a scenario where two vectors have the same or exact opposite direction. When vectors are parallel, there are significant implications in vector algebra. A critical aspect to note is that the cross product of two parallel vectors, as seen in the exercise, results in the zero vector, \( \vec{0} \).

Understanding parallelism is essential when analyzing vector cross product outcomes. Two vectors \( \vec{A} \) and \( \vec{B} \) are parallel if there is a scalar \( k \) such that \( \vec{A} = k \vec{B} \). In the context of the regular hexagon in the exercise, recognizing the parallel vectors is crucial. Since \( \vec{PQ} \) and \( \vec{RS} \) are parallel, their cross product will yield the zero vector, which is a fundamental result of vector parallelism and a significant point for the students to grasp when studying vector operations.

Vector Algebra

Vector algebra encompasses operations performed on vectors within a vector space, including addition, scalar multiplication, dot product, and cross product among others. It provides a framework for solving geometric problems in physics and engineering through mathematical expressions. When dealing with vector cross product, as highlighted in the exercise, this operation takes two vectors and results in a third vector that is perpendicular to both of the original vectors and has a magnitude equal to the area of the parallelogram that the two vectors span.

The cross product is denoted as \( \vec{A} \times \vec{B} \) and the resulting vector's direction is determined by the right-hand rule. In the given exercise, understanding the cross product helps in analyzing vector relationships within the regular hexagon and solving for the non-zero vector resulting from the cross product of \( \vec{PQ} \) with the sum of vectors \( \vec{RS} + \vec{ST} \).

It is essential for mastering vector algebra to comprehend not only the computational aspects but also the geometric interpretations and the conditions under which certain results, such as a zero vector result from parallel vectors, occur. This is particularly relevant in enhancing the student's grasp of how vector operations translate into the geometrical representation of shapes like hexagons.