Problem 91 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(\vec{a}=\hat{i}-\hat{k}, \bar{b}=x \hat{i}+\hat{j}+(1-x) \hat{k}\) and \(\vec{c}=y \hat{i}+x \hat{j}+(1+x-y) \hat{k}\). Then \([\vec{a}, \bar{b}, \vec{c}]\) depends on \(\\{2005]\) (a) only y (b) only \(\mathrm{x}\) (c) both \(\mathrm{x}\) and \(\mathrm{y}\) (d) neither \(\mathrm{x}\) nor \(\mathrm{y}\)

Step-by-Step Solution

Answer

(b) only \(x\).

Step 1: Understand the Scalar Triple Product

The scalar triple product of vectors \(\vec{a}, \vec{b},\) and \(\vec{c}\) is given by \( [\vec{a}, \vec{b}, \vec{c}] = \vec{a} \cdot (\vec{b} \times \vec{c})\). This results in a scalar value, and is computed using the determinant of a matrix formed by these vectors.

Step 2: Set Up the Determinant

To find \([\vec{a}, \vec{b}, \vec{c}]\), express the scalar triple product as the determinant of the matrix formed by \(\vec{a}, \vec{b},\) and \(\vec{c}\): \[\text{Det} \begin{bmatrix} 1 & 0 & -1 \ x & 1 & 1-x \ y & x & 1+x-y \end{bmatrix}\].

Step 3: Evaluate the Determinant

Compute the determinant by expanding along the first row: \[= 1 \cdot \begin{vmatrix} 1 & 1-x \ x & 1+x-y \end{vmatrix} - 0 \cdot \begin{vmatrix} x & 1-x \ y & 1+x-y \end{vmatrix} - 1 \cdot \begin{vmatrix} x & 1 \ y & x \end{vmatrix}\] Evaluating these 2x2 matrices gives: \[= 1((1)(1+x-y) - (x)(1-x)) - (x(1) - 1(y))\] \[= (1 + x - y - x^2 + x) - (x - y)\].

Step 4: Simplify the Expression

Simplify the result:\[= 1 + x - y - x^2 + x - x + y = 1 - x^2 \].

Step 5: Analyze the Dependency

The expression \(1 - x^2\) only involves \(x\) and does not include \(y\). Therefore, \([\vec{a}, \vec{b}, \vec{c}]\) depends only on \(x\).

Key Concepts

Determinant Evaluation

Understanding the process of evaluating a determinant is essential in vector algebra, especially when working with vectors in 3D spaces. A determinant is a unique number that can be calculated from a square matrix. For vectors in 3D, the determinant helps in finding the scalar triple product, which is a measure of the volume of the parallelepiped formed by the vectors. In the given exercise, the scalar triple product is represented as \[\text{Det} \begin{bmatrix}1 & 0 & -1 \x & 1 & 1-x \y & x & 1+x-y \end{bmatrix}\]To evaluate this determinant, you use the method of cofactor expansion, often along a row or a column to simplify the computation.

• Each element in the selected row or column is multiplied by the determinant of the smaller, 2x2 matrix formed by deleting the row and column of that element.
• The signs are alternated, following the checkerboard pattern of plus and minus.

This step-by-step approach not only calculates the determinant but also helps understand the dependency of any variables involved in the matrix on the scalar product.

Vectors in 3D

Vectors are fundamental in physics and mathematics, describing quantities that have both magnitude and direction. In 3D space, vectors extend into another dimension, allowing representation through three components often aligned with the axes: i, j, and k. Vectors in 3D space can be written in the form: \(\vec{v} = a \hat{i} + b \hat{j} + c \hat{k}\) where \(a\), \(b\), and \(c\) are the vector's components. In the original exercise:

• \(\vec{a} = \hat{i} - \hat{k}\)
• \(\vec{b} = x \hat{i} + \hat{j} + (1-x) \hat{k}\)
• \(\vec{c} = y \hat{i} + x \hat{j} + (1+x-y) \hat{k}\)

Each vector is broken down into its components along the x, y, and z axes. This representation allows for easy manipulation and application of vector operations such as addition, subtraction, and cross product, both of which are critical for solving the problem of finding the scalar triple product.

Vector Algebra

Vector algebra is a mathematical system aimed at dealing with the relationships between vectors. This allows the manipulation and understanding of vector quantities through various operations, including:

• Vector Addition: Combining vectors component-wise.
• Scalar Multiplication: Changing the magnitude of a vector by a scalar value.
• Dot Product: Producing a scalar through the multiplication of corresponding components of two vectors.
• Cross Product: Generating a vector perpendicular to two given vectors in 3D space.

The scalar triple product, like in the exercise, involves using both the cross and dot products to determine the volume of a parallelepiped, formed by the vectors \(\vec{a}, \vec{b}, \vec{c}\). This quantity depends on the determinant calculated from the vectors' components. For instance, the computation of \([\vec{a}, \vec{b}, \vec{c}] = \vec{a} \cdot (\vec{b} \times \vec{c})\) is translated into a determinant solution. This captures the essence of vector algebra's power in solving spatial problems. By learning these operations, students can master how to work with vectors in complex 3D environments.