Problem 95 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(\bar{a}=\hat{i}+\hat{j}+\hat{k}, \bar{b}=\hat{i}-\hat{j}+2 \hat{k}\) and \(\bar{c}=x \hat{i}+(x-2) \hat{j}-\hat{k} .\) If the vector \(\bar{c}\) lies in the plane of \(\bar{a}\) and \(\bar{b}\), then \(x\) equals [2007] (A) 0 (B) 1 (C) \(-4\) (D) \(-2\)

Step-by-Step Solution

Answer

The value of \(x\) is \(-2\) (Option D).

Step 1: Understanding Plane Criterion

To determine if \(\bar{c}\) lies in the plane of \(\bar{a}\) and \(\bar{b}\), we need the vectors to be coplanar. This condition can be verified using the scalar triple product equal to zero: \( \bar{a} \cdot (\bar{b} \times \bar{c}) = 0 \).

Step 2: Compute Cross Product \(\bar{b} \times \bar{c}\)

The cross product \(\bar{b} \times \bar{c}\) is:\[\bar{b} \times \bar{c} = \left| \begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \1 & -1 & 2 \x & (x-2) & -1 \\end{array} \right|\]Using determinant expansion: \[ \bar{b} \times \bar{c} = \hat{i}((-1)(-1) - 2(x-2)) - \hat{j}(1(-1) - 2x) + \hat{k}(1(x-2) - (-(x))) \] Simplifying, we find:\[ \bar{b} \times \bar{c} = \hat{i}(1 - 2x + 4) - \hat{j}(-1 - 2x) + \hat{k}(x - 2 + x) \] Resulting in: \[ (5 - 2x)\hat{i} + (1 + 2x)\hat{j} + (2x - 2)\hat{k} \]

Step 3: Compute Scalar Triple Product \(\bar{a} \cdot (\bar{b} \times \bar{c})\)

We need \( \bar{a} \cdot (\bar{b} \times \bar{c}) = 0 \). Thus:\[ \bar{a} \cdot ((5 - 2x)\hat{i} + (1 + 2x)\hat{j} + (2x - 2)\hat{k}) = 0 \]\[ = (1)(5 - 2x) + (1)(1 + 2x) + (1)(2x - 2) \]Simplifying:\[ = 5 - 2x + 1 + 2x + 2x - 2 \]\[ = 4 + 2x \]

Step 4: Solve for \(x\)

Set the expression from Step 3 equal to zero:\[ 4 + 2x = 0 \]Solve for \(x\):\[ 2x = -4 \]\[ x = -2 \]

Key Concepts

Coplanarity

Coplanarity is a condition where three or more vectors lie in the same plane. When vectors are coplanar, you can think of them all as sitting flatly on a tabletop, unable to "jump" out of it. For vectors \( \bar{a}, \bar{b}, \text{ and } \bar{c} \) to be coplanar, a key criterion must be met: the scalar triple product must equal zero. This simple test tells us that if \( \bar{a} \cdot (\bar{b} \times \bar{c}) = 0 \), indeed they are coplanar. Coplanarity is an essential characteristic for determining relationships between vectors, such as verifying if a vector is within the same dimensional space as others.

Scalar Triple Product

The scalar triple product involves three vectors and combines both the dot product and the cross product. It is expressed by the formula \( \bar{a} \cdot (\bar{b} \times \bar{c}) \). Here, the order of operation is crucial: you first take the cross product of two vectors, \( \bar{b} \times \bar{c} \), and then take the dot product of the result with the third vector, \( \bar{a} \). The scalar triple product provides a single scalar value.

• If the value is zero, the vectors are coplanar.
• If not zero, the vectors make up a volume in 3D space.

This product also reflects the volume of the parallelepiped formed by the three vectors, offering a rich geometric insight.

Cross Product

The cross product is a binary operation on two vectors resulting in another vector that is perpendicular to the plane containing the original vectors. This perpendicular vector reflects key properties like magnitude and direction. The magnitude of this resultant vector equals the area of the parallelogram formed by the initial vectors.

Determinant Representation

The cross product \( \bar{b} \times \bar{c} \) can be handily determined using the determinant of a matrix: \[ \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 1 & -1 & 2 \ x & (x-2) & -1 \end{vmatrix} \] This determinant expansion is synonymous with calculating a 3x3 determinant, a process pivotal in understanding spatial vector arrangements.

Determinant Expansion

Determinant expansion is a method used to calculate the determinant of a matrix, often employed in defining the cross product and solving systems of linear equations. The determinant of a 3x3 matrix containing vectors gives us a compact representation of the calculation. In the context of vector operations, it simplifies obtaining the values required for cross products.
The process involves selecting a row or column, then calculating the cofactors — a combination of the remaining two-by-two minor and the sign associated with that position (using the checkerboard pattern of "+" and "-"). This helps express complex calculations in a manageable form, deeply connected to geometric insights such as the vector's perpendicular relation to its initial components.

Mathematical Proofs

Mathematical proofs provide a rigorous method to verify mathematical statements, establishing certainty through logical deduction. To solve the problem from the original exercise, we relied on proof by computation. Each step involved calculated arguments founded in vector operations: verifying coplanarity using scalar triple products, calculating cross products, and methodical determinant expansion.
In mathematics, we often start with an assumption or known fact, proceeding logically through calculations and reasoning until we reach a conclusion. Proofs are not merely for solving equations but building a framework to understand and trust the results. They enable us to ensure validity not only for the problem at hand but across similar scenarios, architecting a consistent mathematical landscape.