Problem 16 - A Complete Resource Book in Mathematics for JEE Main

Question

Let the unit vectors \(a\) and \(b\) be perpendicular to each other and the unit vector \(c\) be inclined at an angle \(\theta\) to both \(a\) and \(b\). If \(c=x a+y b+z(a \times b)\), then (A) \(x=\cos \theta, y=\sin \theta, z=\cos 2 \theta\) (B) \(x=\sin \theta, y=\cos \theta, z=-\cos 2 \theta\) (C) \(x=y=\cos \theta, z^{2}=\cos 2 \theta\) (D) \(x=y=\cos \theta, z^{2}=-\cos 2 \theta\)

Step-by-Step Solution

Answer

Option (D) \(x = y = \cos \theta\), \(z^2 = -\cos 2\theta\).

Step 1: Understand the Vector Relation

The vector \(c\) is expressed in terms of the unit vectors \(a\), \(b\), and their cross product \(a \times b\). Since \(a\) and \(b\) are perpendicular unit vectors, \(a \times b\) is also a unit vector perpendicular to both \(a\) and \(b\). We need to find the coefficients \(x\), \(y\), and \(z\) such that the vector \(c\) is inclined at an angle \(\theta\) to both \(a\) and \(b\).

Step 2: Establish Dot Product Equations

We use the fact that \(c\) makes an angle \(\theta\) with both \(a\) and \(b\). For \(c\) with \(a\): \[c \cdot a = (x a + y b + z (a \times b)) \cdot a = x = \cos \theta\]For \(c\) with \(b\): \[c \cdot b = (x a + y b + z (a \times b)) \cdot b = y = \cos \theta\]

Step 3: Calculate Magnitude of Vector c

Since \(c\) is a unit vector, its magnitude is 1: \[||c|| = \sqrt{x^2 + y^2 + z^2} = 1\]Substitute \(x = \cos \theta\) and \(y = \cos \theta\):\[1 = \sqrt{\cos^2 \theta + \cos^2 \theta + z^2} = \sqrt{2\cos^2 \theta + z^2}\]\[1 = \sqrt{2\cos^2 \theta + z^2}\]

Step 4: Solve for z Using Magnitude Equation

Square both sides to eliminate the square root:\[1 = 2 \cos^2 \theta + z^2\]Rearrange to find \(z^2\):\[z^2 = 1 - 2 \cos^2 \theta\] Simplify using the identity \(\cos(2\theta) = 2\cos^2(\theta) - 1\):\[z^2 = -\cos(2\theta)\]

Step 5: Match the Solution to the Given Options

From the identities and calculations we derived: \(x = y = \cos \theta\) and \(z^2 = -\cos 2\theta\). Compare these with the available options. We find that answer (D) corresponds to \(x = y = \cos \theta\) and \(z^2 = -\cos 2\theta\).

Key Concepts

Dot Product

The dot product is a fundamental operation involving two vectors. It results in a scalar value and provides information about the angle between the vectors. When dealing with unit vectors like in our problem, their magnitudes are always 1.
This makes calculations simpler and clearer.The formula for the dot product of two vectors \(\mathbf{u} = \langle u_1, u_2, u_3 \rangle \) and \(\mathbf{v} = \langle v_1, v_2, v_3 \rangle \) is:

• \(\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3 \)

However, in terms of angles, it is also expressed as:

• \(\mathbf{u} \cdot \mathbf{v} = ||\mathbf{u}|| \cdot ||\mathbf{v}|| \cdot \cos \theta\)

In our problem, this translates to \(x = \cos \theta\) and \(y = \cos \theta\) for the angles between unit vectors \(a\), \(b\), and vector \(c\), since \( ||a|| = ||b|| = ||c|| = 1\).
This relates directly to the inclination angle \(\theta\). By setting these equalities, we extract essential information about how the components of vector \(c\) align with \(a\) and \(b\).

Unit Vector

A unit vector is a vector with a magnitude of 1, which serves as a directional indicator in vector mathematics.
In this problem, the vectors \(a\), \(b\), and \(c\) are all specified as unit vectors and therefore have unit lengths.This characteristic simplifies many calculations as it eliminates the need to normalize vectors, allowing us to focus on just the direction.
In vector operations, such as the equation \( ||c|| = \sqrt{x^2 + y^2 + z^2} = 1 \), involving unit vectors makes it possible to substitute directly without further simplification due to their inherent dimensionless quality.Unit vectors are key when expressing vector directions, making them central in fields like physics and engineering, where directional quantities are critical.
For instance, unit vectors \(a\), \(b\), and their cross product \(a \times b\) help express complex vector relationships effectively. This forms the backbone of problems involving vector projections and transformations, as seen here.

Cross Product

The cross product is a vector product operation that results in another vector. This resulting vector is perpendicular to the original two, making it unique compared to the scalar result of the dot product.
In three-dimensional space, the cross product of two vectors \(\mathbf{u} = \langle u_1, u_2, u_3 \rangle \) and \(\mathbf{v} = \langle v_1, v_2, v_3 \rangle \) is:

• \(\mathbf{u} \times \mathbf{v} = \langle u_2v_3 - u_3v_2, u_3v_1 - u_1v_3, u_1v_2 - u_2v_1 \rangle \)

The magnitude of the cross product is given by:

• \(||\mathbf{u} \times \mathbf{v}|| = ||\mathbf{u}|| \cdot ||\mathbf{v}|| \cdot \sin \theta\)

Applying this to unit vectors \(a\) and \(b\) yields another unit vector, \(a \times b\), perpendicular to both \(a\) and \(b\). In our exercise, the vector \(c\) is expressed using \(a\), \(b\), and \(a \times b\).
This demonstrates how the cross product can expand a vector space basis, accommodating components perpendicular to the existing vectors.

Angle Between Vectors

Understanding the angle between vectors is vital when determining how vectors align or diverge in space.
The angle \(\theta\) indicates the degree of inclination between any two vectors, which can help solve for unknown components, vector magnitudes, or further trigonometric relationships.For two vectors \(\mathbf{u}\) and \(\mathbf{v}\), the angle \(\theta\) can be found using the dot product formula:

• \(\cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{||\mathbf{u}|| \cdot ||\mathbf{v}||}\)

Since our vectors are unit vectors, we have \(||\mathbf{u}|| = ||\mathbf{v}|| = 1\).
Thus, this simplifies our calculations to simply evaluating the dot product. In the context of the exercise, setting the angles \(\theta\) between \(a\), \(b\), and \(c\) solidifies the solution, ensuring that \(c\)'s components \(x\) and \(y\) align perpendicularly to \(a\) and \(b\). This aligns with the expression for \(c\), confirming vector inclination within the spatial framework described.