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

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

Option C, \( \hat{i} + \hat{j} - 2 \hat{k} \), is the correct vector \( \vec{b} \).

Step 1: Understand the given vectors

We have two vectors given: \( \vec{a} = \vec{j} - \vec{k} \) and \( \vec{c} = \vec{i} - \vec{j} - \vec{k} \). We need a vector \( \vec{b} \) such that \( \vec{a} \times \vec{b} + \vec{c} = \vec{0} \) and \( \vec{a} \cdot \vec{b} = 3 \).

Step 2: Solve the equation involving cross product

From \( \vec{a} \times \vec{b} + \vec{c} = \vec{0} \), we can express it as \( \vec{b} = - (\vec{a} \times \vec{b}) \). Calculating \( \vec{a} \times \vec{b} \), assume \( \vec{b} = x \vec{i} + y \vec{j} + z \vec{k} \). The determinant for cross product \( \vec{a} \times \vec{b} \) becomes \([-y\vec{i} - x\vec{j} + x\vec{k}]\). This transforms the given equation to: \(-y\vec{i} - x\vec{j} + x\vec{k} = - (\vec{i} - \vec{j} - \vec{k})\).

Step 3: Equate components from vectors

From the equation \(-y\vec{i} - x\vec{j} + x\vec{k} = -\vec{i} + \vec{j} + \vec{k}\), equate the components, giving: 1) \(-y = -1 \Rightarrow y = 1\)2) \(-x = 1 \Rightarrow x = -1\)3) \(x = 1 \Rightarrow x = 1\).

Step 4: Verify conditions

Verify with \( \vec{a} \cdot \vec{b} = 3 \) using \( \vec{b} = \vec{i} + \vec{j} - 2\vec{k}\). Calculate \( \vec{a} \cdot \vec{b} = (0\cdot 1) + (1\cdot 1) + (-1\cdot -2) = 0 + 1 + 2 = 3 \). Hence, this \( \vec{b} \) satisfies the conditions.

Key Concepts

Cross Product

The cross product, also known as the vector product, is an operation between two vectors in three-dimensional space. It results in another vector that is perpendicular to both of the original vectors. This perpendicularity is a crucial feature of the cross product.
To compute the cross product of two vectors, we use the following determinant formula:
For vectors \( \vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k} \) and \( \vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k} \), the cross product \( \vec{a} \times \vec{b} \) is given by:
\[\vec{a} \times \vec{b} = \begin{vmatrix}\hat{i} & \hat{j} & \hat{k} \a_1 & a_2 & a_3 \b_1 & b_2 & b_3\end{vmatrix}\]
This matrix expands to yield:
\[(a_2b_3 - a_3b_2)\hat{i} - (a_1b_3 - a_3b_1)\hat{j} + (a_1b_2 - a_2b_1)\hat{k}\]
In the context of our exercise, finding \( \vec{b} \) involves setting up an equation based on the cross product and solving for unknown components of \( \vec{b} \). After expressing \( \vec{a} \times \vec{b} \), we equate it as given in the problem's condition to solve for these components. It is essential to maintain the the negative sign correctly as it impacts all components involved.

Dot Product

The dot product, sometimes referred to as the scalar product, combines two vectors to produce a scalar quantity. This operation is based on the geometric concept of projecting one vector onto another.
The formula for the dot product of two vectors \( \vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k} \) and \( \vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k} \) is:
\[\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3\]
This product gives us a single number, not a vector.
The importance of the dot product lies in its ability to provide a measure of the extent to which two vectors point in the same direction. A dot product of zero indicates that the vectors are orthogonal, or at right angles, to each other.
In the given exercise, once the components of \( \vec{b} \) are proposed, the condition \( \vec{a} \cdot \vec{b} = 3 \) verifies if the chosen \( \vec{b} \) is correct, checking through the computations to ensure they satisfy this equation.

Vector Equations

Vector equations represent relationships between vectors. These are often expressed in terms of vector addition, scalar multiplication, and operations such as dot and cross products.
Equations involving vectors can combine different vector operations. For instance, in the given problem, the equation \( \vec{a} \times \vec{b} + \vec{c} = \vec{0} \) involves both vector addition and a cross product. This same equation is solved by rearranging it to find the unknown vector \( \vec{b} \).
Key steps in working with vector equations include:

• Identify which operations are used in the equation.
• Rearrange the equation to isolate the unknown vector if possible.
• Solve using known vector algebra, verifying with provided conditions.

In problems with multiple conditions, such as this one with a cross product and a separate dot product, solutions are validated against all given requirements to ensure accuracy.