Problem 34 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem Statement

We are given vectors, with properties describing their relationships, and asked to determine the absolute value of the dot product of a cross product of two vectors, with a third vector. Specifically, we need to evaluate how this value compares to given fractions.

Step 2: Express the Given Equation

Start with the equation given: \[ w + (w \times u) = v \] Rearrange it to express \( w \):\[ w = v - (w \times u) \]

Step 3: Square Each Side and Simplify

Since we want to find \((u \times v) \cdot w\), first find the magnitude of each side of the equation. Note that \(w\), \(u\), and \(v\) are unit vectors, so with magnitudes equal to 1, the expression involving dot product and cross product simplification is used to find other relations.

Step 4: Cross Product and Dot Product Relations

Using the given vectors' properties, express \((u \times v) \cdot (v - (w \times u))\) to further simplify the main sought expression. This provides connectivity between dot and cross products.

Step 5: Solve for Required Expression

Use vector identity properties, such as the distributive property to solve for the required expression based on the values of \((u \times v)\) and the derived form of \(w\).

Step 6: Evaluate the Result Against Choices

With \((u \times v) \cdot w\) having been computed and simplified, compare its magnitude to the given options: \(\frac{1}{3}, \frac{1}{2}, <, \leq , >, \geq\). Analyze correctly to interpret whether the result lies within the boundaries of the given choices.

Key Concepts

Cross Product

The cross product is a particular operation that happens between two vectors in three-dimensional space. For two vectors \( \mathbf{a} \) and \( \mathbf{b} \), the cross product \( \mathbf{a} \times \mathbf{b} \) results in a new vector that is perpendicular to both \( \mathbf{a} \) and \( \mathbf{b} \). This unique property makes it extremely useful in physics and engineering to find orthogonal vectors.

Some essential characteristics of the cross product include:

• The magnitude or length of \( \mathbf{a} \times \mathbf{b} \) is given by the formula \( |\mathbf{a}||\mathbf{b}| \sin(\theta) \), where \( \theta \) is the angle between the two vectors.
• The direction is determined by the right-hand rule, which states that if you point your index finger in the direction of \( \mathbf{a} \) and your middle finger in the direction of \( \mathbf{b} \), your thumb points in the direction of the cross product.

When considering unit vectors, the principle remains the same but simplifies calculations. Since the magnitude of unit vectors is 1, the cross product of two orthogonal unit vectors is another unit vector.

Dot Product

Let's dive into the dot product, another fundamental operation in vector algebra. The dot product of two vectors \( \mathbf{a} \cdot \mathbf{b} \) yields a scalar quantity, not a vector. It's calculated as \( |\mathbf{a}||\mathbf{b}| \cos(\theta) \), where \( \theta \) is again the angle between the two vectors.

Key features of the dot product include:

• It provides a measure of how much one vector extends in the direction of another.
• It will be zero if the vectors are perpendicular.

In the context of this exercise, where \((u \times v) \cdot w\) is evaluated, the dot product plays a pivotal role. The problem relies on you understanding that the dot product will provide the projection of vector \( w \) along the direction of \( u \times v \), impacting its comparison with given fractions like \( \frac{1}{3} \) and \( \frac{1}{2} \).

Unit Vectors

Unit vectors are essential building blocks in vector algebra. These are vectors that have a magnitude of exactly 1 and are used to specify directions. The most common unit vectors used in coordinate systems are \( \mathbf{i}, \mathbf{j}, \) and \( \mathbf{k} \), which align with the x, y, and z axes respectively.

Some critical points to remember about unit vectors include:

• They have no units of length or size attached, only direction.
• In the calculation and operations with vectors, unit vectors simplify many expressions, especially when vector properties like cross and dot products are involved.

In our given problem, the vectors \( u \) and \( v \) are unit vectors. This implies \( |u| = 1 \) and \( |v| = 1 \). These properties help simplify expressions because they remove the need for repeated magnitude calculations, streamlining the evaluation of operations like the dot product in the context given.