Problem 104 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(\hat{a}\) and \(\hat{b}\) be two unit vectors. If the vectors \(\vec{c}=\hat{a}+2 \hat{b}\) and \(d=5 \hat{a}-4 \hat{b}\) are perpendicular to each other, then the angle between \(\hat{a}\) and \(\hat{b}\) is (A) \(\frac{\pi}{6}\) (B) \(\frac{\pi}{2}\) (C) \(\frac{\pi}{3}\) (D) \(\frac{\pi}{4}\)

Step-by-Step Solution

Answer

The angle between \(\hat{a}\) and \(\hat{b}\) is \(\frac{\pi}{3}\). (C)

Step 1: Understanding the Problem

We need to determine the angle between two unit vectors \( \hat{a} \) and \( \hat{b} \). The vectors \( \vec{c} = \hat{a} + 2 \hat{b} \) and \( \vec{d} = 5 \hat{a} - 4 \hat{b} \) are given to be perpendicular. This means their dot product is zero.

Step 2: Write the Dot Product Equation

Using the condition that vectors \( \vec{c} \) and \( \vec{d} \) are perpendicular, we have the equation:\[ \vec{c} \cdot \vec{d} = (\hat{a} + 2\hat{b}) \cdot (5\hat{a} - 4\hat{b}) = 0 \] which results in \[ \hat{a}\cdot(5\hat{a} - 4\hat{b}) + 2\hat{b}\cdot(5\hat{a} - 4\hat{b}) = 0 \].

Step 3: Expand the Dot Product Equation

Expand each dot product using the distributive property:\[ 5(\hat{a} \cdot \hat{a}) - 4(\hat{a} \cdot \hat{b}) + 10(\hat{b} \cdot \hat{a}) - 8(\hat{b} \cdot \hat{b}) = 0 \].

Step 4: Simplify the Dot Product Terms

Since \( \hat{a} \) and \( \hat{b} \) are unit vectors, \( \hat{a} \cdot \hat{a} = 1 \) and \( \hat{b} \cdot \hat{b} = 1 \). Also, \( \hat{a} \cdot \hat{b} = \hat{b} \cdot \hat{a} = \cos \theta \). Substitute these in the equation: \[ 5(1) - 4(\cos \theta) + 10(\cos \theta) - 8(1) = 0 \].

Step 5: Solve for Cosine of the Angle

Simplify the equation from Step 4:\[ 5 - 8 + (10 - 4)\cos \theta = 0 \]\[ -3 + 6\cos \theta = 0 \]\[ 6 \cos \theta = 3 \]\[ \cos \theta = \frac{1}{2} \].

Step 6: Determine the Angle

Since \( \cos \theta = \frac{1}{2} \), this corresponds to an angle \( \theta = \frac{\pi}{3} \) (60 degrees).

Key Concepts

Unit Vectors

Unit vectors are fundamental components in vector mathematics. They have a magnitude of one and are used to indicate direction in a coordinate system. In any vector space, the concept of a unit vector can be applied. Unit vectors are often represented with a "hat" symbol, like \( \hat{a} \) or \( \hat{b} \). Here are some key points about unit vectors:

• They have a length of 1, which makes calculations simpler.
• They can be used to build other vectors by scaling them.
• In physics and engineering, unit vectors provide a way to express directions without concerning themselves with magnitude.

In our exercise, both \( \hat{a} \) and \( \hat{b} \) are unit vectors, which simplifies solving for the angle since their respective dot products will yield values based on direction alone.

Dot Product

The dot product is a fundamental operation in vector algebra that computes a single number from two vectors. This scalar value is essential in determining angles between vectors and their relative orientations. Understanding how to calculate the dot product is key to solving many vector-related problems, including our exercise where vectors are perpendicular.

• The dot product of two vectors \( \mathbf{u} = (u_1, u_2, ..., u_n) \) and \( \mathbf{v} = (v_1, v_2, ..., v_n) \) is given by \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + ... + u_nv_n \).
• If the dot product is zero, the vectors are perpendicular to one another.
• Dot products can be used to calculate the angle \( \theta \) between two vectors, where \( \cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}||\mathbf{v}|} \).

In the step-by-step solution, the dot product of the vectors \( \vec{c} \) and \( \vec{d} \) being zero confirms they are perpendicular, allowing us to find the angle between \( \hat{a} \) and \( \hat{b} \).

Vector Algebra

Vector algebra is the study of vectors, objects that have both magnitude and direction. It involves different operations and rules that help us manipulate vectors to solve various mathematical and physical problems. A few key operations include addition, subtraction, and multiplication (dot and cross products).

• Vector addition and subtraction involve combining components: \( \mathbf{u} + \mathbf{v} = (u_1 + v_1, u_2 + v_2, ...) \).
• The dot product is a type of multiplication that results in a scalar, while the cross product results in another vector.
• The vector equation solutions give insight into geometrical and physical interpretations of phenomena.

This exercise used vector algebra by equating vector expressions and using their dot product to find relationships between them. It demonstrated how vector components \( \hat{a} \) and \( \hat{b} \) combine and how their operations unravel needed vector properties.

Perpendicular Vectors

When vectors are perpendicular, their dot product equals zero. This property is crucial because it helps identify right angle relationships between two vector directions. Perpendicular vectors form the basis of orthogonal vector sets, which have applications in both simple geometry and complex multidimensional spaces.

• Two vectors \( \mathbf{u} \) and \( \mathbf{v} \) being perpendicular implies \( \mathbf{u} \cdot \mathbf{v} = 0 \).
• They provide insights into angles; if the dot product equals zero, the vectors form a 90-degree angle.
• Perpendicularity is essential in vector projections and in finding vector bases.

In the problem, \( \vec{c} \) and \( \vec{d} \) are perpendicular. This property was key to finding the angle between \( \hat{a} \) and \( \hat{b} \), illustrating the zero dot product leading to the calculated cosine value.