Problem 57 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \(\vec{a}=2 \hat{i}+\hat{j}+2 \hat{k}\), then the value of \(|\hat{i} \times(\vec{a} \times \hat{i})|^{2}+|\hat{j} \times(\vec{a} \times \hat{j})|^{2}+|\hat{k} \times(\vec{a} \times \hat{k})|^{2}\) is equal to

Step-by-Step Solution

Answer

The value is 18.

Step 1: Understand the Problem

We need to calculate the magnitude squares of triple cross products using the vector \(\vec{a} = 2 \hat{i} + \hat{j} + 2 \hat{k}\) with each of the unit vectors \(\hat{i}, \hat{j}, \hat{k}\). Then, sum up these values.

Step 2: Evaluate \(\hat{i}\) Cross Product

Find \(\hat{i} \times (\vec{a} \times \hat{i})\). Use the vector triple product identity: \(\mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = (\mathbf{A} \cdot \mathbf{C})\mathbf{B} - (\mathbf{A} \cdot \mathbf{B})\mathbf{C}\). Thus: \[\hat{i} \times (\vec{a} \times \hat{i}) = (\hat{i} \cdot \hat{i})\vec{a} - (\hat{i} \cdot \vec{a})\hat{i} = \vec{a} - 2\hat{i} = \hat{j} + 2\hat{k}.\]

Step 3: Calculate Magnitude Square for \(\hat{i}\)

Calculate \(|\hat{i} \times (\vec{a} \times \hat{i})|^2|\): \[|(\hat{j} + 2\hat{k})|^2 = 1^2 + 2^2 = 1 + 4 = 5.\]

Step 4: Evaluate \(\hat{j}\) Cross Product

Find \(\hat{j} \times (\vec{a} \times \hat{j})\) using the identity: \[\hat{j} \times (\vec{a} \times \hat{j}) = (\hat{j} \cdot \hat{j})\vec{a} - (\hat{j} \cdot \vec{a})\hat{j} = \vec{a} - \hat{j} = 2\hat{i} + 2\hat{k}.\]

Step 5: Calculate Magnitude Square for \(\hat{j}\)

Calculate \(|\hat{j} \times (\vec{a} \times \hat{j})|^2\): \[|(2\hat{i} + 2\hat{k})|^2 = 2^2 + 2^2 = 4 + 4 = 8.\]

Step 6: Evaluate \(\hat{k}\) Cross Product

Find \(\hat{k} \times (\vec{a} \times \hat{k})\) using the identity: \[\hat{k} \times (\vec{a} \times \hat{k}) = (\hat{k} \cdot \hat{k})\vec{a} - (\hat{k} \cdot \vec{a})\hat{k} = \vec{a} - 2\hat{k} = 2\hat{i} + \hat{j}.\]

Step 7: Calculate Magnitude Square for \(\hat{k}\)

Calculate \(|\hat{k} \times (\vec{a} \times \hat{k})|^2\): \[|(2\hat{i} + \hat{j})|^2 = 2^2 + 1^2 = 4 + 1 = 5.\]

Step 8: Sum the Magnitude Squares

Add up all the computed squared magnitudes: \[|\hat{i} \times (\vec{a} \times \hat{i})|^2 + |\hat{j} \times (\vec{a} \times \hat{j})|^2 + |\hat{k} \times (\vec{a} \times \hat{k})|^2 = 5 + 8 + 5 = 18.\]

Key Concepts

Triple Cross Product

The triple cross product, also known as the vector triple product, involves two cross products performed sequentially. It is given by the expression \( \mathbf{A} \times (\mathbf{B} \times \mathbf{C}) \). Understanding the vector triple product identity simplifies calculations:

• \( \mathbf{A} \cdot \mathbf{B} \) denotes the dot product, which is a scalar output.
• The identity—\( \mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = (\mathbf{A} \cdot \mathbf{C})\mathbf{B} - (\mathbf{A} \cdot \mathbf{B})\mathbf{C} \)—helps in avoiding complicated direct cross product calculations.

When you apply this identity, it allows you to break down the expression into simpler vector additions and scalars, making complex product calculations far more manageable. This becomes very handy when evaluating algebraic expressions with vectors in any dimension.

Magnitude Square

The magnitude square of a vector \( \mathbf{v} \) is calculated as the sum of the squares of its components. It’s represented as \( |\mathbf{v}|^2 \).

• In a three-dimensional space, if \( \mathbf{v} = a\hat{i} + b\hat{j} + c\hat{k} \), the magnitude square is \( a^2 + b^2 + c^2 \).
• Calculating the magnitude square is crucial for determining the length of a resultant vector without having to square root any components.

The magnitude square provides a simple way to compare vector sizes or to sum up the overall magnitude of multiple vectors derived from cross products. It holds great importance in problems where finding explicit magnitude is unnecessary, especially in vector algebra where direct magnitude comparisons are more insightful.

Vector Operations

Vector operations such as addition, scalar multiplication, dot product, and cross product are the building blocks of vector algebra. Here’s a quick overview:

• Addition: Vectors are added component-wise, such as \( \mathbf{v}_1 + \mathbf{v}_2 = (a_1 + a_2)\hat{i} + (b_1 + b_2)\hat{j} + (c_1 + c_2)\hat{k} \).
• Scalar Multiplication: This scales a vector by multiplying each of its components by a scalar, \( k \), so \( k\mathbf{v} = ka\hat{i} + kb\hat{j} + kc\hat{k} \).
• Dot Product: This yields a scalar and is calculated as \( \mathbf{v}_1 \cdot \mathbf{v}_2 = a_1a_2 + b_1b_2 + c_1c_2 \).
• Cross Product: Produces a vector that is perpendicular to the original two vectors, \( \mathbf{v}_1 \times \mathbf{v}_2 \).

Mastering these operations allows you to combine and analyze vectors efficiently, which is essential in breaking down complex problems in physics and engineering.

Unit Vectors

Unit vectors are fundamental in vector algebra, representing direction. Each unit vector has a magnitude of one, which makes them useful for defining directions along coordinate axes.

• Standard Unit Vectors: In three-dimensional space, they include \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \).
• Normalization: Any vector can be converted into a unit vector, ensuring it maintains direction but scales to a magnitude of one. This is done by dividing the vector by its magnitude.
• Direction Representation: Unit vectors are ideal for representing direction only without affecting magnitude.

In exercises involving multiple vectors, such as the one described, unit vectors help break down a vector into manageable components aligned with the axes, simplifying mathematical operations and physical interpretations.