Problem 10 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(a\) and \(b\) are two unit vectors, then the vector \((a+b)\) \(\times(a \times b)\) is parallel to the vector (A) \(a-b\) (B) \(a+b\) (C) \(2 a-b\) (D) \(2 a+b\)

Step-by-Step Solution

Answer

The vector is parallel to option (C) \(2a - b\).

Step 1: Understanding Unit Vectors

Unit vectors are vectors with a magnitude of 1. If \(a\) and \(b\) are unit vectors, then their dot products are simpler, i.e., \(a \cdot a = 1\) and \(b \cdot b = 1\).

Step 2: Recall Cross Product Properties

Two important properties are: 1) The cross product of two vectors \( (c \times d) \) is a vector perpendicular to both \(c\) and \(d\). 2) The formula for triple vectors: \( (u + v) \times (u \times v) = (u \cdot v) u - (u \cdot u) v \).

Step 3: Apply Triple Product Identity

By applying the identity, we obtain: \((a+b) \times (a \times b) = ([a \cdot b]a - [a \cdot a]b) + ([b \cdot b]a - [b \cdot a]b)\).

Step 4: Simplify the Expanded Expression

Since \(a \cdot a = 1\) and \(b \cdot b = 1\), and \(a \cdot b\) (a scalar), simplify: \([a \cdot b]a - b + a - [a \cdot b]b \). Thus, the result is \(2a - b\).

Step 5: Determine the Parallel Vector

The simplified result is \(2a - b\), which directly matches option (C). Therefore, the vector \((a+b) \times (a \times b)\) is parallel to \(2a - b\).

Key Concepts

Unit Vectors

Unit vectors are essential building blocks in vector algebra. Think of them as the vectors with a special status—because their magnitude is always 1. Magnitude is just another word for length or size of the vector.
Whenever you see a unit vector like \( a \) or \( b \), you know right away that \( a \cdot a = 1 \) and \( b \cdot b = 1 \) because the dot product of a vector with itself is just the square of its magnitude. This makes calculations involving unit vectors simpler, as you often work with these perfect square magnitudes.

• Why they're important: Unit vectors provide direction without scaling the quantities they're used with.
• Common examples: In 3D space, the standard unit vectors are \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \) representing the x, y, and z axes respectively.

Understanding unit vectors enhances your ability to simplify vector expressions and solve problems efficiently.

Cross Product Properties

The cross product is an operation between two vectors which results in a third vector that is perpendicular to the plane of the first two. Imagine two vectors \( c \) and \( d \) lying on a flat plane like a sheet of paper. The cross product \( c \times d \) gives you a vector that sticks out of the paper!
Here are some key properties of the cross product:

• Perpendicularity: The result \( c \times d \) is always perpendicular to both \( c \) and \( d \).
• Magnitude: The magnitude of \( c \times d \) is equal to the area of the parallelogram that the vectors span, calculated as \( |c||d|\sin(\theta) \), where \( \theta \) is the angle between \( c \) and \( d \).
• Non-commutative: This operation is not commutative, meaning \( c \times d eq d \times c \), in fact, \( c \times d = - (d \times c) \).

These properties make the cross product a unique and crucial operation in vector algebra, especially in physics applications like torque and angular momentum.

Triple Product Identity

The triple product identity is a formula that simplifies the handling of the vector cross product when three vectors are involved. It's useful when you need to expand expressions like \((u + v) \times (u \times v)\).
Here's the triple product identity formula:\[(u + v) \times (u \times v) = (u \cdot v) u - (u \cdot u) v\]This powerful identity allows you to break down complex vector expressions into simpler terms that are easier to work with.

• Utilization: It's most beneficial when solving vector equations as seen in problems involving torque or magnetic forces in physics, where cross products are prevalent.
• Practical Example: In the given exercise, applying the triple product identity helped transform \((a+b) \times (a \times b)\) into a simpler form to determine the parallel vector.

Grasping the triple product identity aids in tackling vector problems more efficiently and comprehensively.