Let `hata and hatb` two unit vectors. If the vectors `c = hata +2hatb and d = 5hata - 4hatb` are perpendicular to each other , then the angle btween `hata and hatb` is

Options

A. `pi/6`
B. `pi/2`
C. `pi/3`
D. `pi/4`

Answer

To solve the problem step by step, we need to find the angle between the two unit vectors \( \hat{a} \) and \( \hat{b} \) given that the vectors \( \mathbf{c} = \hat{a} + 2\hat{b} \) and \( \mathbf{d} = 5\hat{a} - 4\hat{b} \) are perpendicular.

### Step 1: Understand the condition of perpendicularity
Two vectors are perpendicular if their dot product is zero. Therefore, we need to set up the equation:
\[ \mathbf{c} \cdot \mathbf{d} = 0 \]

### Step 2: Compute the dot product of \( \mathbf{c} \) and \( \mathbf{d} \)
Substituting the expressions for \( \mathbf{c} \) and \( \mathbf{d} \):
\[ \mathbf{c} \cdot \mathbf{d} = (\hat{a} + 2\hat{b}) \cdot (5\hat{a} - 4\hat{b}) \]
Using the distributive property of the dot product:
\[ \mathbf{c} \cdot \mathbf{d} = \hat{a} \cdot (5\hat{a}) + \hat{a} \cdot (-4\hat{b}) + 2\hat{b} \cdot (5\hat{a}) + 2\hat{b} \cdot (-4\hat{b}) \]
This simplifies to:
\[ = 5(\hat{a} \cdot \hat{a}) - 4(\hat{a} \cdot \hat{b}) + 10(\hat{b} \cdot \hat{a}) - 8(\hat{b} \cdot \hat{b}) \]

### Step 3: Substitute known values
Since \( \hat{a} \) and \( \hat{b} \) are unit vectors:
\[ \hat{a} \cdot \hat{a} = 1 \quad \text{and} \quad \hat{b} \cdot \hat{b} = 1 \]
Thus, we have:
\[ = 5(1) - 4(\hat{a} \cdot \hat{b}) + 10(\hat{a} \cdot \hat{b}) - 8(1) \]
This simplifies to:
\[ = 5 - 8 + 6(\hat{a} \cdot \hat{b}) = -3 + 6(\hat{a} \cdot \hat{b}) \]

### Step 4: Set the dot product to zero
Setting the dot product equal to zero gives:
\[ -3 + 6(\hat{a} \cdot \hat{b}) = 0 \]
Solving for \( \hat{a} \cdot \hat{b} \):
\[ 6(\hat{a} \cdot \hat{b}) = 3 \implies \hat{a} \cdot \hat{b} = \frac{1}{2} \]

### Step 5: Relate the dot product to the angle
The dot product of two vectors can also be expressed in terms of the angle \( \theta \) between them:
\[ \hat{a} \cdot \hat{b} = |\hat{a}| |\hat{b}| \cos \theta \]
Since both \( \hat{a} \) and \( \hat{b} \) are unit vectors:
\[ \hat{a} \cdot \hat{b} = 1 \cdot 1 \cdot \cos \theta = \cos \theta \]
Thus, we have:
\[ \cos \theta = \frac{1}{2} \]

### Step 6: Find the angle \( \theta \)
The angle \( \theta \) whose cosine is \( \frac{1}{2} \) is:
\[ \theta = \cos^{-1}\left(\frac{1}{2}\right) = 60^\circ \quad \text{or} \quad \frac{\pi}{3} \text{ radians} \]

### Final Answer
The angle between the vectors \( \hat{a} \) and \( \hat{b} \) is \( 60^\circ \) or \( \frac{\pi}{3} \).

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