Problem 100 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If the vectors \(\vec{c}, \vec{a}=x \hat{i}+y \hat{j}+z \hat{k}\) and \(\hat{b}=\hat{j}\) are such that \(\vec{a}, \vec{c}\) and \(\vec{b}\) form a right handed system then \(\vec{c}\) is: [2002] (a) \(z \hat{i}-x \hat{k}\) (b) \(\overrightarrow{0}\) (c) \(\hat{y j}\) (d) \(-z \hat{i}+x \hat{k}\)

Step-by-Step Solution

Answer

Option (d): \(-z \hat{i} + x \hat{k}\).

Step 1: Understand the Right-Handed System

A "right-handed system" of vectors means that the cross product of two vectors in the system follows the right-hand rule. This means if \(\vec{a} \times \vec{b} = \vec{c}\), \(\vec{c}\) should point in the direction of your right thumb when your fingers point from \(\vec{a}\) to \(\vec{b}\).

Step 2: Apply Cross Product Formula

Given \(\vec{a} = x \hat{i} + y \hat{j} + z \hat{k}\) and \(\vec{b} = \hat{j}\), use the cross product formula: \(\vec{a} \times \vec{b} = (x \hat{i} + y \hat{j} + z \hat{k}) \times \hat{j}\).

Step 3: Calculate the Cross Product

\(\vec{a} \times \hat{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ x & y & z \ 0 & 1 & 0 \end{vmatrix} = \hat{i}(y\cdot 0 - 1\cdot z) - \hat{j}(x\cdot 0 - z\cdot 0) + \hat{k}(x\cdot 1 - 0\cdot 0) = -z \hat{i} + x \hat{k}\)

Step 4: Compare with Given Options

The vector obtained from the cross product, \(-z \hat{i} + x \hat{k}\), corresponds exactly with option (d).

Key Concepts

Cross Product

The cross product is a fundamental concept in vector algebra that calculates a vector perpendicular to two given vectors in three-dimensional space. The cross product of two vectors, say \( \vec{a} \) and \( \vec{b} \), results in a new vector \( \vec{c} \). This vector \( \vec{c} \) is orthogonal to both \( \vec{a} \) and \( \vec{b} \). For vectors expressed in terms of the unit vectors \( \hat{i}, \hat{j}, \) and \( \hat{k} \), you utilize the determinant formula to find the cross product:
\[\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \end{vmatrix}\]
This results in a vector whose components are calculated using the minor of each unit vector. For instance, when you find \( \vec{a} \times \hat{b} = -z \hat{i} + x \hat{k} \), this implies that the new vector \( \vec{c} \) obtained is perpendicular to both \( \vec{a} \) and \( \vec{b} \).
Essentially, you'll derive the cross product by calculating the determinant as a shortcut to obtaining the coordinates of the new vector.

Right-Hand Rule

The right-hand rule is a standard mnemonic used in physics to determine the direction of the cross product vector. When working with vectors in a three-dimensional space, this rule ensures that you choose the correct direction for the resultant vector. Here’s how you can apply it:

• Position your right hand so that your fingers align with the first vector, \( \vec{a} \).
• Curl your fingers towards the direction of the second vector, \( \vec{b} \).
• Your thumb, which is perpendicular to your curled fingers, will point in the direction of the resultant vector \( \vec{c} \).

This technique is incredibly useful as it intuitively builds on defining a "right-handed" system. In this exercise, when you compute \( \vec{a} \times \vec{b} \), applying the right-hand rule confirms that the direction matches the vector \( -z \hat{i} + x \hat{k} \), confirming it fits with option (d).
The right-hand rule not only helps understand vector direction but also solidifies the concept of handedness in vector systems.

Vector Algebra

Vector algebra is the study of mathematical operations involving vectors, which are quantities characterized by magnitude and direction. In this field, vectors are typically expressed in Cartesian coordinates using unit vectors \( \hat{i}, \hat{j}, \hat{k} \) which correspond to the x, y, and z-axis respectively.
In vector calculations, you encounter operations such as addition, scalar multiplication, dot products, and cross products. Each operation follows specific rules and provides different insights into the vector quantities. For cross products, the critical aspect is to find a vector that is normal to the plane formed by two input vectors, using determinants as discussed previously.
When dealing with vector operations, especially cross products, it is important to maintain clarity between different vector-based operations by:

• Identifying vectors numerically using Cartesian components.
• Using unit vectors consistently for clear direction indication.
• Applying algebraic rules comprehensively to ensure accurate results.

The problem discussed exemplifies these principles, leading to a deeper understanding of how vector algebra aids in solving physics and engineering problems.