Problem 104 - A Complete Resource Book in Mathematics for JEE Main 2017

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 [2012] (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 \(\frac{\pi}{3}\).

Step 1: Identify Perpendicular Condition

Given two vectors \(\vec{c} = \hat{a} + 2 \hat{b}\) and \(\vec{d} = 5 \hat{a} - 4 \hat{b}\), it is stated that they are perpendicular. This means their dot product equals zero. Let's set this condition as the starting point for solving the problem.

Step 2: Calculate the Dot Product

Calculate the dot product of the vectors \(\vec{c}\) and \(\vec{d}\): \( (\hat{a} + 2 \hat{b}) \cdot (5 \hat{a} - 4 \hat{b}) = \hat{a} \cdot 5 \hat{a} - \hat{a} \cdot 4 \hat{b} + 2 \hat{b} \cdot 5 \hat{a} - 2 \hat{b} \cdot 4 \hat{b} \). Simplifying, we get: \(5 \hat{a} \cdot \hat{a} - 4 \hat{a} \cdot \hat{b} + 10 \hat{b} \cdot \hat{a} - 8 \hat{b} \cdot \hat{b}\).

Step 3: Apply Unit Vector Properties

Recall that for any unit vector \(\hat{v}\), \(\hat{v} \cdot \hat{v} = 1\) and \(\hat{v}\) is perpendicular to itself only. Use these properties to simplify the dot product:\(5(1) - 4(\hat{a} \cdot \hat{b}) + 10(\hat{b} \cdot \hat{a}) - 8(1) = 0\). This reduces to: \(5 + 6 \hat{a} \cdot \hat{b} - 8 = 0\).

Step 4: Solve for \(\hat{a} \cdot \hat{b}\)

Simplify the equation from the previous step to isolate \(\hat{a} \cdot \hat{b}\): \(-3 + 6 \hat{a} \cdot \hat{b} = 0\) \makes it: \(6 \hat{a} \cdot \hat{b} = 3\) \thus \(\hat{a} \cdot \hat{b} = \frac{1}{2}\).

Step 5: Determine the Angle Between Vectors

The dot product can also be expressed as \(\hat{a} \cdot \hat{b} = \cos \theta\), where \(\theta\) is the angle between the vectors.Thus, \(\cos \theta = \frac{1}{2}\). From trigonometric tables, we know that \(\theta = \frac{\pi}{3}\). Thus, the angle between \(\hat{a}\) and \(\hat{b}\) is \(\frac{\pi}{3}\).

Key Concepts

Vectors

Vectors are fundamental in mathematics and physics. They are used to represent quantities that have both a magnitude and a direction. For instance, a vector can represent forces like wind or the direction and speed of a moving object. You visualize vectors as arrows. The length of the arrow reflects the vector's magnitude, while the direction of the arrow shows the direction of the vector.

In mathematical notation, vectors are often denoted with an arrow over a letter or as bold letters, such as \( \vec{v} \) or simply **v**. A vector can be absolute, showing a certain length and direction, or described as a unit vector, which simply indicates direction and has a magnitude of one.

When adding or subtracting vectors, you can think of it as moving along directions in multiple dimensions. For example, adding **v** and **w** involves placing the tail of **w** at the head of **v** and drawing the resultant vector from the tail of **v** to the head of **w**. Mathematics allows these operations to be efficiently performed using component form, where vectors are broken down into their components along the coordinate axes.

Dot Product

The dot product, also known as the scalar product, is a fundamental operation involving two vectors. It results in a scalar rather than another vector. The formula for the dot product of two vectors \(\vec{a}\) and \(\vec{b}\) is:
\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta \]
where \(\theta\) is the angle between the vectors. This expression shows that the dot product combines the magnitudes of the vectors with the cosine of the angle between them. A common application of the dot product is determining if two vectors are perpendicular (orthogonal). If the dot product is zero, the vectors are orthogonal as the angle between them is 90 degrees, and therefore, \(\cos(90^\circ) = 0\).

In component form, the dot product of vectors \(\vec{a} = (a_1, a_2, a_3)\) and \(\vec{b} = (b_1, b_2, b_3)\) is computed as:
\[ \vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3 \]
Understanding the dot product is crucial in solving problems involving angles between vectors, projections, and energy calculations in physics.

Unit Vectors

Unit vectors are an essential concept in vector mathematics. A unit vector is a vector of length one and is used to describe direction. Denoted typically by a hat, such as \(\hat{a}\), unit vectors help standardize the direction without concern for magnitude. They are crucial in transforming vector magnitudes into directions. For example, any vector \( \vec{v} \) can be turned into a unit vector by dividing each component by its magnitude:
\[ \hat{v} = \frac{\vec{v}}{|\vec{v}|} \]
where \( |\vec{v}| \) is the magnitude of the vector \( \vec{v} \).

Unit vectors in a coordinate system are typically represented by the standard unit vectors \( \hat{i}, \hat{j}, \hat{k} \), which align with the x, y, and z axes respectively. These unit vectors are widely used in mathematical physics and engineering fields to express vector quantities more effectively. They help in simplifying complex vector calculations by reducing the vector into its direction, allowing you to focus on computation of angles and projections easily.