Problem 80 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Statement 1: The vectors \(\vec{a}, \vec{b}\) and \(\vec{c}\) lie in the same plane if and only if \(\vec{a} \cdot(\vec{b} \times \vec{c})=0\) \(\Rightarrow\) Statement \(2:\) The vectors \(u\) and \(v\) are perpendicular if and only if \(\vec{u} \cdot \vec{v}=0\) where \(\vec{u} \times \vec{v}\) is a vector perpendicular to the plane of \(\vec{u}\) and \(\vec{v}\). [Online May 26, 2012] (a) Statement 1 is false, Statement 2 is true. (b) Statement 1 is true, Statement 2 is true, Statement 2 is correct explanation for Statement 1 . (c) Statement 1 is true, Statement 2 is false. (d) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1 .

Step-by-Step Solution

Answer

(d) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.

Step 1: Understanding Statement 1

Statement 1 states that vectors \( \vec{a}, \vec{b}, \text{ and } \vec{c} \) lie in the same plane if and only if the scalar triple product, \( \vec{a} \cdot (\vec{b} \times \vec{c}) = 0 \), is valid. This is a well-known mathematical fact; the scalar triple product is zero if and only if the vectors are coplanar, meaning they lie within the same two-dimensional plane in three-dimensional space.

Step 2: Evaluating Statement 1

Based on mathematical properties, Statement 1 is true. The scalar triple product can indeed be used to determine if three vectors are coplanar.

Step 3: Understanding Statement 2

Statement 2 describes perpendicular vectors \( \vec{u} \) and \( \vec{v} \) with \( \vec{u} \cdot \vec{v} = 0 \), implying a zero dot product. This is a basic vector property: when the dot product is zero, vectors are perpendicular. It also mentions \( \vec{u} \times \vec{v} \), which is the cross product yielding a vector perpendicular to the plane formed by \( \vec{u} \) and \( \vec{v} \).

Step 4: Evaluating Statement 2

Statement 2 is true as it correctly describes the properties of perpendicular vectors and outlines the geometric significance of both the dot and cross products.

Step 5: Analyzing Explanation Relationship

Although both statements are true, Statement 2 does not directly explain Statement 1. The zero dot product condition in Statement 2 does not clarify why the scalar triple product being zero implies coplanarity.

Key Concepts

Scalar Triple Product

The scalar triple product is a crucial concept in vector operations. It involves three vectors: let’s say \( \vec{a}, \vec{b}, \text{ and } \vec{c} \). This product is represented by \( \vec{a} \cdot (\vec{b} \times \vec{c}) \). Here’s how it works:

• The cross product \( \vec{b} \times \vec{c} \) results in a vector perpendicular to both \( \vec{b} \) and \( \vec{c} \).
• The dot product \( \vec{a} \cdot (\vec{b} \times \vec{c}) \) computes the magnitude of \( \vec{a} \) projected in the direction of \( \vec{b} \times \vec{c} \). If they are perpendicular, this magnitude is zero.

If this scalar triple product equals zero, it confirms that the three vectors are coplanar. This means they all lie in the same plane. When vectors are coplanar, their scalar triple product ultimately indicates their dependency on one another.

Coplanarity

Coplanarity is a term used in vector mathematics to describe when three or more vectors lie in the same geometric plane. Verifying coplanarity is essential because it helps us understand the spatial relationship between the vectors.
The condition for coplanarity with three vectors, \( \vec{a}, \vec{b}, \text{ and } \vec{c} \), can be determined using the scalar triple product:

• If \( \vec{a} \cdot (\vec{b} \times \vec{c}) = 0 \), the vectors are coplanar.

This means no vector can be uniquely defined in terms of the others. Practically, this computes to a scenario where all three vectors lie flat on the same plane in three-dimensional space.

Dot Product

The dot product, an essential operation in vector algebra, provides a measure of how much two vectors align with each other. Let’s consider two vectors \( \vec{u} \) and \( \vec{v} \). The dot product is calculated as:

• \( \vec{u} \cdot \vec{v} = \| \vec{u} \| \cdot \| \vec{v} \| \cdot \cos \theta \), where \( \theta \) is the angle between the vectors.

The result is a scalar quantity.
When the dot product equals zero, \( \vec{u} \) and \( \vec{v} \) are perpendicular. This means the angle between them is 90 degrees, and thus, they have no projection onto one another. Understanding the dot product can help visualize how vectors relate to each other in terms of direction and orientation.

Cross Product

The cross product, unlike the dot product, results in a vector. When finding the cross product of two vectors \( \vec{u} \) and \( \vec{v} \), the outcome vector is perpendicular to the plane formed by \( \vec{u} \) and \( \vec{v} \). Here's how it is useful:

• The formula is \( \vec{u} \times \vec{v} = \| \vec{u} \| \cdot \| \vec{v} \| \cdot \sin \theta \cdot \hat{n} \), where \( \hat{n} \) is the unit vector perpendicular to both \( \vec{u} \) and \( \vec{v} \).

This operation is useful for determining the orientation of vectors in space.
Since the result is a vector, it provides not just the magnitude of separation in terms of physical distance, but also the specific axis of rotation or push it exerts in three-dimensional space. For example, if you imagine rotational forces, the cross product can model the pivot movement efficiently.