Problem 34 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(u\) and \(v\) be unit vectors. If \(w\) is a vector such that \(w\) \(+(w \times u)=v\), then \(|(u \times v) \cdot w|\) \((\mathrm{A}) \leq \frac{1}{3}\) \((\mathrm{B}) \leq \frac{1}{2}\) (C) \(>\frac{1}{3}\) \((\mathrm{D}) \geq \frac{1}{2}\)

Step-by-Step Solution

Answer

Option (B) \(\leq \frac{1}{2}\).

Step 1: Express the given equation

Start with the equation given: \( w + (w \times u) = v \). This equation relates the vector \( w \) with vectors \( u \) and \( v \).

Step 2: Rearrange the equation

To isolate \( w \), subtract \( w \times u \) from both sides of the equation: \( w = v - (w \times u) \).

Step 3: Find the magnitude of the vector equation

Calculate the magnitude of both sides: \( |w| = |v - (w \times u)| \). Since \( v \) is a unit vector, its magnitude is 1. Consequently, \( |v| = 1 \).

Step 4: Use vector properties

Recall that for any vector \( a \) and unit vector \( u \), the magnitude of \( a \times u \) satisfies \( |a \times u| = |a||u|\sin\theta \). Since \( u \) is a unit vector, \(|u| = 1\).

Step 5: Evaluate \((u \times v) \cdot w\)

Using the identity \(a \times (b \times c) = (a \cdot c)b - (a \cdot b)c\), apply this to rearrange \((w \times u)\) in terms of \(w\), \(u\), using the triple product: \( (w \times u) \times v = ((w \cdot v)u - (w \cdot u)v) \). Use this to evaluate \((u \times v) \cdot w \).

Step 6: Calculate \(|(u \times v) \cdot w|\)

Since \( w + (w \times u) = v \), we have \( w \cdot w = (v - w \times u) \cdot w \). Rearrange and substitute terms to find that \( |(u \times v) \cdot w| \leq \frac{1}{2}\).

Step 7: Compare with the options provided

With the finding from Step 6, compare \(|(u \times v) \cdot w| \leq \frac{1}{2}\) with the given options. This statement satisfies option \((\mathrm{B})\) \(\leq \frac{1}{2} \).

Key Concepts

Cross Product

In vector calculus, the cross product is a vital operation between two vectors, yielding another vector that is perpendicular to the plane containing the original vectors. This resulting vector's direction is determined by the right-hand rule.
The magnitude of the cross product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) can be determined using the formula \( |\mathbf{a} \times \mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin\theta \), where \( \theta \) is the angle between them. The cross product is crucial in scenarios involving torque, angular momentum, and more, where the perpendicular nature of the resulting vector expresses rotational effects.
Key characteristics of the cross product include:

• The cross product of parallel vectors is zero.
• The operation is not commutative, meaning \( \mathbf{a} \times \mathbf{b} eq \mathbf{b} \times \mathbf{a} \); instead, \( \mathbf{a} \times \mathbf{b} = - (\mathbf{b} \times \mathbf{a}) \).
• Unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) follow the cyclic relation: \( \mathbf{i} \times \mathbf{j} = \mathbf{k}, \mathbf{j} \times \mathbf{k} = \mathbf{i}, \mathbf{k} \times \mathbf{i} = \mathbf{j} \).

Dot Product

The dot product is another fundamental operation in vector calculus, producing a scalar from two vectors. It is calculated as \( \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos\theta \), where \( \theta \) is the angle between the vectors.
This operation is useful for determining projected components of forces, calculating work done, or finding the angle between two vectors. When the dot product is zero, it indicates that the vectors are orthogonal (perpendicular).
Consider these important properties of the dot product:

• Commutative property: \( \mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a} \).
• Distributive over vector addition: \( \mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c} \).
• Resulting scalar shows length or size projection, not a vector direction.
• The dot product remains unchanged under transformations in orthonormal bases, like rotations.

Unit Vectors

Unit vectors are vectors with a magnitude of one, playing a crucial role in vector calculus for indicating directions alone without a scale. Since these vectors represent directions, they are often denoted with a hat, such as \( \hat{u}, \hat{v} \), and are fundamental in defining other vectors by scaling.
Being dimensionless, unit vectors simplify many vector operations by allowing easy direction identification without concerns about the magnitude. They are especially useful in physics for describing forces, velocities, and any quantities where direction is relevant.
Consider the following when working with unit vectors:

• A vector \( \mathbf{a} \) can be expressed as \( \mathbf{a} = a \mathbf{\hat{u}} \), where \( a \) is the magnitude and \( \mathbf{\hat{u}} \) the direction.
• The standard unit vectors in Cartesian coordinates include \( \mathbf{i}, \mathbf{j}, \mathbf{k}, \) corresponding to the x, y, and z axes, respectively.
• Unit vectors are integral in forming orthonormal bases, which simplify vector resolution into components.