Problem 101 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \(\vec{a}, \vec{b}, \vec{c}\) are vectors such that \([\vec{a} \vec{b} \vec{c}]=4\) then \([\vec{a} \times \vec{b} \vec{b} \times \vec{c} \vec{c} \times \vec{a}]=\) (a) 16 (b) 64 (c) 4 (d) 8

Step-by-Step Solution

Answer

16

Step 1: Understanding the Cube Volume Formula

In vector algebra, the expression \([\vec{a} \vec{b} \vec{c} ]\) denotes the scalar triple product \(\vec{a} \cdot (\vec{b} \times \vec{c})\), which represents the volume of the parallelepiped formed by the vectors \(\vec{a}, \vec{b}, \vec{c}\). Given \([\vec{a} \vec{b} \vec{c} ] = 4\), the volume is 4.

Step 2: Evaluating the Parallelepiped Volume using Cross Products

We are asked to evaluate the scalar triple product \([\vec{a} \times \vec{b} \vec{b} \times \vec{c} \vec{c} \times \vec{a} ]\). We can use the identity \((\vec{a} \times \vec{b}) \cdot (\vec{b} \times \vec{c}) \times (\vec{c} \times \vec{a}) = ([\vec{a} \vec{b} \vec{c}])^2\) to solve this, which calculates the volume squared of the original vectors.

Step 3: Applying the Identity Correction

The identity states that \((\vec{a} \times \vec{b}) \cdot (\vec{b} \times \vec{c}) \times (\vec{c} \times \vec{a}) = ([\vec{a} \vec{b} \vec{c}])^2\). Given this, we substitute with the provided volume \([\vec{a} \vec{b} \vec{c}] = 4\).

Step 4: Calculate the Result

Plug in \([\vec{a} \vec{b} \vec{c}]=4\) into the identity: \(([\vec{a} \vec{b} \vec{c}])^2 = 4^2 = 16\). Thus, \([\vec{a} \times \vec{b} \vec{b} \times \vec{c} \vec{c} \times \vec{a} ] = 16.\)

Key Concepts

Scalar Triple Product

The scalar triple product of three vectors, \(\vec{a}, \vec{b}, \vec{c}\), is denoted by \([\vec{a} \vec{b} \vec{c} ]\). It is defined as \(\vec{a} \cdot (\vec{b} \times \vec{c})\). This expression not only involves both dot and cross products but also represents a significant geometric interpretation.
The scalar triple product gives the volume of the parallelepiped (a kind of 3D box) formed by these vectors. If you imagine placing the vectors like edges extending from a common vertex, \([\vec{a} \vec{b} \vec{c} ]\) tells you how much space is enclosed within. For instance, if \([\vec{a} \vec{b} \vec{c} ]=4\), the volume of the parallelepiped is 4 cubic units.

• The scalar triple product provides insight into the alignment of vectors. If it's zero, the vectors are coplanar, meaning they lie in the same plane and don't span any volume.
• Mathematically, the scalar triple product is invariant under any cyclic permutation of the vectors, meaning \([\vec{a} \vec{b} \vec{c} ] = [\vec{b} \vec{c} \vec{a} ] = [\vec{c} \vec{a} \vec{b} ]\).

Cross Product Identities

Cross product identities are essential tools in vector algebra that assist in simplifying expressions involving multiple vector cross products. The identity used here is:

• \( (\vec{a} \times \vec{b}) \cdot (\vec{b} \times \vec{c}) \times (\vec{c} \times \vec{a}) = ([\vec{a}, \vec{b}, \vec{c}])^2 \)

This identity involves the square of the scalar triple product and it is very useful in evaluating certain types of vector products. In the given exercise, this identity allowed converting a complex-looking cross product expression into a simple scalar multiplication.
Cross products themselves are essential as they yield vectors perpendicular to the original input vectors, effectively giving us insights into the orientation of spatial planes made by these vectors. They follow distributive and anticommutative properties, meaning:

• \( \vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c} \)
• \( \vec{a} \times \vec{b} = - (\vec{b} \times \vec{a}) \)

These foundational properties simplify calculations in various areas, such as physics and computer graphics, by ensuring predictable behavior of vector operations.

Parallelepiped Volume

The volume of a parallelepiped is a fundamental concept in vector algebra, allowing us to find the space enclosed by three vectors. As discussed, the volume is given by the scalar triple product, \([\vec{a} \vec{b} \vec{c}]\). However, when analyzing complex structures or expressions, understanding how this translates with different vector arrangements is important.
In the context of the exercise, understanding how the identity \((\vec{a} \times \vec{b}) \cdot (\vec{b} \times \vec{c}) \times (\vec{c} \times \vec{a}) = ([\vec{a}, \vec{b}, \vec{c}])^2\) was applied reveals the hidden layers of vector arrangement possibilities and their effects on volume. It shows that even if vectors are rearranged through cross products, the calculated volume squared is still consistent when this specific identity holds.
Parallelepiped volumes often crop up in physics and engineering when calculating real-world scenarios involving forces and movement through space. Comprehension of this geometric interpretation of vector quantities is invaluable for resolving spatial problems effectively.