Problem 89 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(a, b\) and \(c\) be distinct non-negative numbers. If the vectors \(a \hat{i}+a \hat{j}+c \hat{k}, \hat{i}+\hat{k}\) and \(c \hat{i}+c \hat{j}+b \hat{k}\) lie in a plane, then \(c\) is \([2005]\) (A) the Geometric Mean of \(a\) and \(b\) (B) the Arithmetic Mean of \(a\) and \(b\) (C) equal to zero (D) the Harmonic Mean of \(a\) and \(b\)

Step-by-Step Solution

Answer

(D) The Harmonic Mean of \(a\) and \(b\).

Step 1: Understand the problem

We are given three vectors in a 3D space: \( \vec{v_1} = a \hat{i} + a \hat{j} + c \hat{k} \), \( \vec{v_2} = \hat{i} + \hat{k} \), and \( \vec{v_3} = c \hat{i} + c \hat{j} + b \hat{k} \). These vectors lie in a plane, meaning they are linearly dependent.

Step 2: Identify the condition for linear dependence

Vectors \( \vec{v_1} \), \( \vec{v_2} \), and \( \vec{v_3} \) are linearly dependent if there exist scalars \( \lambda, \mu, \) not all zero, such that \( \lambda \vec{v_1} + \mu \vec{v_2} + u \vec{v_3} = \vec{0} \). This is equivalent to their cross product being zero.

Step 3: Calculate the cross product \( \vec{v_1} \times \vec{v_2} \)

The cross product \( \vec{v_1} \times \vec{v_2} \) can be calculated as follows:\[\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ a & a & c \ 1 & 0 & 1 \end{vmatrix} = (a \cdot 1 - 0 \cdot c) \hat{i} - (a \cdot 1 - c \cdot 1) \hat{j} + (a \cdot 0 - a \cdot 1) \hat{k}\] = a\hat{i} - (a - c)\hat{j} - a\hat{k}.

Step 4: Calculate the cross product \( (\vec{v_1} \times \vec{v_2}) \times \vec{v_3} \)

Take the cross product from Step 3: \( a\hat{i} - (a - c)\hat{j} - a\hat{k} \). Compute its cross product with \( \vec{v_3} = c \hat{i} + c \hat{j} + b \hat{k} \).\[\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ a & -(a-c) & -a \ c & c & b \end{vmatrix} = ((a-c)b + ac)\hat{i} - (ab + ac)\hat{j} + (-a - c^2)\hat{k}.\] Upon simplification set this resulting determinant to zero.

Step 5: Solve the equation obtained from the determinant

We obtained the vector \(((a-c)b + ac)\hat{i} - (ab + ac)\hat{j} + (-a - c^2)\hat{k}\). For this to hold all components must equate to zero. Thus, we solve: \[ ((a-c)b + ac) = 0 \] and \[ ab + ac = 0 \]. Simplifying, we find \[ c(a+b-ac-bc) = 0 \]. Since \(ceq 0\) and \(a, b, c\) are non-negative distinct values, equate \(a+b = ac+bc\) leading to \(c = \frac{ab}{a+b} \).

Step 6: Interpret the result

This expression \( c = \frac{ab}{a+b} \) is the formula for the harmonic mean of \(a\) and \(b\). Hence, we conclude that among the given options, \(c\) is the harmonic mean of \(a\) and \(b\).

Key Concepts

Linear Dependence of Vectors

In a 3D space, vectors are said to be linearly dependent if you can express one vector as a combination of others. To put it simply, if three vectors lie in the same plane, they're likely linked together through linear dependence. This means there are coefficients for these vectors that sum up to zero, allowing a perfect combination to make them cancel each other out entirely. Here, we check for linear dependence by using the condition \( \lambda \vec{v_1} + \mu \vec{v_2} + u \vec{v_3} = \vec{0} \), where not all of \( \lambda, \mu, \) and \( u \) are zero.
This becomes a useful property in mathematical problems when deciding whether vectors are coplanar. It's like having different strings that, when pulled together in the right way, create a straight line or a flat surface.

Cross Product

The cross product is a crucial tool in vector algebra. It helps determine a vector that's perpendicular to two given vectors. Imagine using the right-hand rule for an easy way to visualize this. Place your index finger along the first vector, your middle finger along the second, and your thumb will point in the direction of the cross product.

In the exercise, we compute the cross product to check if the vectors are linearly dependent. Here's the formula for the cross product of two vectors: \( \vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ A_x & A_y & A_z \ B_x & B_y & B_z \end{vmatrix} \).
This gives us a new vector which, if it's zero, confirms that the original vectors are coplanar. In such cases, the determinant approach provides an effective method for calculating cross products and finding dependencies.

Harmonic Mean

The harmonic mean is one of the means that allows calculation of an average value in specific scenarios. Unlike the arithmetic or geometric mean, the harmonic mean focuses on the reciprocal relationship between numbers.
For two values \( a \) and \( b \), the harmonic mean is given by \( c = \frac{2ab}{a+b} \). This curious relationship flips the focus from simple averages to something more aligned with synchronizing rates or ratios.

It's like thinking of two people paddling a canoe together: the harmonic mean helps imagine the average effect of their combined effort, rather than just their individual speeds. In our exercise, finding \( c \) as the harmonic mean of \( a \) and \( b \) provides the elegant solution that combines their interactions perfectly.