Problem 75 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \([\vec{a} \times \vec{b} \quad \vec{b} \times \vec{c} \quad \vec{c} \times \vec{a}]=\lambda[\vec{a} \vec{b} \vec{c}]^{2}\) then \(\lambda\) is equal to \([2014]\) (a) 0 (b) 1 (c) 2 (d) 3

Step-by-Step Solution

Answer

The value of \(\lambda\) is 0, option (a).

Step 1: Understand the Notation

The notation \([ \vec{a} \quad \vec{b} \quad \vec{c} ]\) represents the scalar triple product, which is equivalent to the determinant of the matrix with vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) as columns. It computes a volume.

Step 2: Expand LHS with Cross Product Properties

The left-hand side \([\vec{a} \times \vec{b} \quad \vec{b} \times \vec{c} \quad \vec{c} \times \vec{a}]\) involves cross products, whose determinant can be solved (by vector identity and determinant properties) as zero since vectors \(\vec{a} \times \vec{b}\), \(\vec{b} \times \vec{c}\), \(\vec{c} \times \vec{a}\) are coplanar with the original vectors \(\vec{a}, \vec{b}, \vec{c}\).

Step 3: Resolve the Right Hand Expression

The right-hand side is \(\lambda [\vec{a} \quad \vec{b} \quad \vec{c}]^{2}\). The term \([\vec{a} \quad \vec{b} \quad \vec{c}]\) is not zero if the vectors are non-coplanar. Thus, \([\vec{a} \quad \vec{b} \quad \vec{c}]^{2} > 0\).

Step 4: Equate the Two Expressions and Solve for \(\lambda\)

Since LHS is 0 and RHS is \(\lambda \times V^2\) (where \(V^2 > 0\)), for the equality to hold, \(\lambda\) must be zero. Therefore, \(0 = \lambda \times V^2\) leads to \(\lambda = 0\).

Step 5: Conclusion

Based on the calculation and reasoning above, \(\lambda\) must be zero for the equation to hold.

Key Concepts

Cross Product

The cross product is a vital operation in vector mathematics, particularly within three-dimensional space. It produces a vector that is perpendicular to two input vectors, thus forming a right triangle with the given vectors. The cross product of vectors \( \vec{a} \) and \( \vec{b} \) is denoted as \( \vec{a} \times \vec{b} \). This product is defined only in three dimensions and is crucial for identifying orthogonal vectors between two lines or planes.
One key property of the cross product is that its magnitude represents the area of the parallelogram formed by the two vectors. Mathematically, this is expressed as:\[\|\vec{a} \times \vec{b}\| = \|\vec{a}\| \|\vec{b}\| \sin(\theta)\] where \( \theta \) is the angle between \( \vec{a} \) and \( \vec{b} \). Therefore, the cross product not only gives the direction but also provides a measure of magnitude equivalent to area.
Some useful properties include:

• Anti-commutative: \( \vec{a} \times \vec{b} = - \vec{b} \times \vec{a} \)
• Distributive over addition: \( \vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c} \)
• Scalar multiplication: \( c (\vec{a} \times \vec{b}) = (c\vec{a}) \times \vec{b} = \vec{a} \times (c\vec{b}) \)

Understanding these principles aids in comprehending more complex geometric relationships, such as in physics or computer graphics.

Determinant Properties

The determinant of a matrix is a scalar attribute that provides insight into numerous geometric and algebraic properties of the linear transformation the matrix represents. Determinants are calculated for square matrices, and they serve several purposes, such as indicating whether a matrix is invertible or the volume scaling factor associated with the linear transformation.
For 3D vectors, the determinant of the matrix constructed by vectors \( \vec{a}, \vec{b}, \vec{c} \) as its columns (or rows) measures the volume of the parallelepiped they form. The scalar triple product \( [\vec{a} \quad \vec{b} \quad \vec{c}] \) is equivalent to this determinant, calculated as:\[\det(\vec{a}, \vec{b}, \vec{c}) = \vec{a} \cdot (\vec{b} \times \vec{c})\]The determinant offers several crucial properties:

• A determinant of zero indicates that the matrix is singular, meaning it cannot be inverted.
• The determinant is linear concerning both rows and columns, facilitating expansion through specific operations or strategies like cofactor expansion.
• Swapping two rows or columns changes the sign of the determinant.

In the context of determining \( \lambda \) in the given problem, recognizing that the left-side determinant equates to zero due to vector coplanarity significantly simplifies resolving the provided equation.

Coplanar Vectors

Vectors are considered coplanar when they all lie within the same plane. In three-dimensional space, any three vectors that do not determine a non-zero volume when used to form a parallelepiped will be coplanar, meaning the scalar triple product of these vectors is zero. This happens when the determinant of the corresponding matrix, as explained earlier, equals zero.
A simple geometric interpretation is that when vectors \( \vec{a}, \vec{b}, \vec{c} \) are coplanar, any of these vectors can be expressed as a linear combination of the others. This can be denoted by the relationship:\[\vec{c} = x \vec{a} + y \vec{b}\]where \( x \) and \( y \) are scalars. Understanding this parameter is critical for solving problems that involve calculating the scalar triple product because when vectors are coplanar, the volume enclosed is zero, and hence, their cross product determinants lead to a zero-based evaluation system. This principle was applied to deduce that \( \lambda \) in the given equation must be zero, ensuring the coherence of the mathematical expression.
Coplanar vectors provide great insight into the dependencies and spatial relationships between vectors in geometric transformation and linear algebra contexts.