Problem 100 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(\vec{a}=\vec{j}-\vec{k}\) and \(\vec{c}=\vec{i}-\vec{j}-\vec{k}\). Then, the vector \(\vec{b}\) satisfying \(\vec{a} \times \vec{b}+\vec{c}=\overrightarrow{0}\) and \(a \cdot \vec{b}=3\) is (A) \(2 \hat{i}-\hat{j}+2 \hat{k}\) (B) \(\hat{i}-\hat{j}-2 \hat{k}\) (C) \(\hat{i}+\hat{j}-2 \hat{k}\) (D) \(-\hat{i}+\hat{j}-2 \hat{k}\)

Step-by-Step Solution

Answer

The correct vector is D: \(-\hat{i} + \hat{j} - 2\hat{k}\).

Step 1: Understanding the Problem

We need to find a vector \( \vec{b} \) such that the vector cross product \( \vec{a} \times \vec{b} \) added to vector \( \vec{c} \) equals zero, i.e., \( \vec{a} \times \vec{b} + \vec{c} = \overrightarrow{0} \), and the dot product of \( \vec{a} \cdot \vec{b} = 3 \).

Step 2: Express the Cross Product Equation

Rearrange the equation \( \vec{a} \times \vec{b} + \vec{c} = \overrightarrow{0} \) to \( \vec{a} \times \vec{b} = -\vec{c} \). Given \( \vec{a} = \vec{j} - \vec{k} \) and \( \vec{c} = \vec{i} - \vec{j} - \vec{k} \), substitute \( \vec{c} \) to get \( \vec{a} \times \vec{b} = - (\vec{i} - \vec{j} - \vec{k}) \) or \( \vec{a} \times \vec{b} = -\vec{i} + \vec{j} + \vec{k} \).

Step 3: Calculate \( \vec{a} \times \vec{b} \)

Use the determinant method for cross products: the determinant of a matrix formed by unit vectors, components of \( \vec{a} \), and components of \( \vec{b} \). Suppose \( \vec{b} = x \vec{i} + y \vec{j} + z \vec{k} \): \[\vec{a} \times \vec{b} = \begin{vmatrix}\hat{i} & \hat{j} & \hat{k} \0 & 1 & -1 \x & y & z\end{vmatrix} = \hat{i}(y - z) - \hat{j}(x) + \hat{k}(0-x)\]Thus, \( \vec{a} \times \vec{b} = (y-z)\hat{i} - x\hat{j} - x\hat{k} \) must equal \( -\vec{i} + \vec{j} + \vec{k} \).

Step 4: Equate Components for Cross Product

From \( \vec{a} \times \vec{b} = -\vec{i} + \vec{j} + \vec{k} \), equate components: \[(y-z) = -1, \quad -x = 1, \quad -x = 1\]Solve to get \( x = -1 \). Substitute \( x = -1 \) in the first equation to find \( y - z = -1 \).

Step 5: Solve Dot Product Condition

The dot product \( \vec{a} \cdot \vec{b} = 3 \) gives:\[(\vec{j} - \vec{k}) \cdot (x \vec{i} + y \vec{j} + z \vec{k}) = 0\cdot x + 1 \cdot y + (-1) \cdot z = y-z\]Using these conditions:1. \( x = -1 \)2. \( y - z = -1 \)3. \( y - z = 3 \)Thus, solving gives \( x = -1, y = 1, z = 2 \).

Step 6: Verify and Select the Correct Option

The vector \( \vec{b} = -1 \vec{i} + 1 \vec{j} + 2 \vec{k} \) matches option D: \(-\hat{i} + \hat{j} - 2\hat{k}\), therefore the correct option is D.

Key Concepts

Cross Product

The cross product, often represented as \( \vec{a} \times \vec{b} \), is a binary operation on two vectors in three-dimensional space, resulting in another vector that is perpendicular to both inputs. One fascinating property of the cross product is that its magnitude equals the area of the parallelogram that the two vectors span.
To find the cross product, you can use the determinant of a matrix constructed with unit vectors \( \hat{i}, \hat{j}, \hat{k} \) in the first row and the components of each vector in the subsequent rows. For example:
\[ \vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \end{vmatrix} \]
Where \( \vec{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \) and \( \vec{b} = b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} \). The result is a vector that:

• Has components calculated as follows: \( \vec{a} \times \vec{b} = (a_2b_3 - a_3b_2)\hat{i} - (a_1b_3 - a_3b_1)\hat{j} + (a_1b_2 - a_2b_1)\hat{k} \).

By observing the cross product, you can gain insight into the geometric relationship between two vectors.

Dot Product

The dot product, symbolized as \( \vec{a} \cdot \vec{b} \), results in a scalar value and provides important geometric insights, such as determining whether two vectors are perpendicular. For two vectors, the dot product blends their magnitudes with the cosine of the angle between them, captured by the formula:
\[ \vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3 \]
This formula emerges from analyzing each component's product and summing them. The result, a single number, reveals:

• When \( \vec{a} \cdot \vec{b} = 0 \), the vectors are orthogonal, hinting at no "shared directionality."
• How much of one vector "extends in the direction of" another, providing nuances about their alignment.

Ultimately, understanding the dot product helps in applications like projecting one vector onto another or analyzing force components in physics.

Component Form

Vectors can break down into smaller, more manageable parts called components. Each component aligns with a spatial axis, typically represented by \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \) for the x, y, and z axes, respectively.
When expressed in component form, a vector crystallizes into a sum of these axis-aligned contributions. For instance, a vector \( \vec{v} \) in component form reads as \( \vec{v} = v_1\hat{i} + v_2\hat{j} + v_3\hat{k} \), where:

• \( v_1 \), \( v_2 \), and \( v_3 \) capture the magnitudes along each axis.
• This setup allows effortless interaction of vectors, aiding calculations like addition, subtraction, and those involving dot or cross products.

Component form acts as a blueprint, unfolding vectors' directional strengths and simplistically solving complex problems like finding forces or determining resultant velocities.