If `a = hati+2hatj-3hatk and b = 3hati-hatj+2hatk` then the angle between the vectors a + b and a - b is

Options

A. `60^@`
B. `90^@`
C. `45^@`
D. `55^@`

Answer

To find the angle between the vectors \( \mathbf{a} + \mathbf{b} \) and \( \mathbf{a} - \mathbf{b} \), we will follow these steps:

### Step 1: Define the vectors
Given:
\[ \mathbf{a} = \hat{i} + 2\hat{j} - 3\hat{k} \]
\[ \mathbf{b} = 3\hat{i} - \hat{j} + 2\hat{k} \]

### Step 2: Calculate \( \mathbf{a} + \mathbf{b} \)
To find \( \mathbf{a} + \mathbf{b} \):
\[ \mathbf{a} + \mathbf{b} = (\hat{i} + 2\hat{j} - 3\hat{k}) + (3\hat{i} - \hat{j} + 2\hat{k}) \]
Combine like terms:
\[ = (1 + 3)\hat{i} + (2 - 1)\hat{j} + (-3 + 2)\hat{k} \]
\[ = 4\hat{i} + 1\hat{j} - 1\hat{k} \]
Thus,
\[ \mathbf{a} + \mathbf{b} = 4\hat{i} + \hat{j} - \hat{k} \]

### Step 3: Calculate \( \mathbf{a} - \mathbf{b} \)
To find \( \mathbf{a} - \mathbf{b} \):
\[ \mathbf{a} - \mathbf{b} = (\hat{i} + 2\hat{j} - 3\hat{k}) - (3\hat{i} - \hat{j} + 2\hat{k}) \]
Combine like terms:
\[ = (1 - 3)\hat{i} + (2 + 1)\hat{j} + (-3 - 2)\hat{k} \]
\[ = -2\hat{i} + 3\hat{j} - 5\hat{k} \]
Thus,
\[ \mathbf{a} - \mathbf{b} = -2\hat{i} + 3\hat{j} - 5\hat{k} \]

### Step 4: Calculate the dot product \( (\mathbf{a} + \mathbf{b}) \cdot (\mathbf{a} - \mathbf{b}) \)
Now, we need to calculate the dot product:
\[ (\mathbf{a} + \mathbf{b}) \cdot (\mathbf{a} - \mathbf{b}) = (4\hat{i} + \hat{j} - \hat{k}) \cdot (-2\hat{i} + 3\hat{j} - 5\hat{k}) \]
Calculating the dot product:
\[ = 4 \cdot (-2) + 1 \cdot 3 + (-1) \cdot (-5) \]
\[ = -8 + 3 + 5 \]
\[ = 0 \]

### Step 5: Calculate the magnitudes
Next, we calculate the magnitudes of \( \mathbf{a} + \mathbf{b} \) and \( \mathbf{a} - \mathbf{b} \):
\[ |\mathbf{a} + \mathbf{b}| = \sqrt{(4)^2 + (1)^2 + (-1)^2} = \sqrt{16 + 1 + 1} = \sqrt{18} \]
\[ |\mathbf{a} - \mathbf{b}| = \sqrt{(-2)^2 + (3)^2 + (-5)^2} = \sqrt{4 + 9 + 25} = \sqrt{38} \]

### Step 6: Use the dot product to find the cosine of the angle
Using the formula for the cosine of the angle \( \theta \):
\[ \cos \theta = \frac{(\mathbf{a} + \mathbf{b}) \cdot (\mathbf{a} - \mathbf{b})}{|\mathbf{a} + \mathbf{b}| \cdot |\mathbf{a} - \mathbf{b}|} \]
Substituting the values:
\[ \cos \theta = \frac{0}{\sqrt{18} \cdot \sqrt{38}} = 0 \]

### Step 7: Find the angle \( \theta \)
Since \( \cos \theta = 0 \), we have:
\[ \theta = 90^\circ \]

### Final Answer
The angle between the vectors \( \mathbf{a} + \mathbf{b} \) and \( \mathbf{a} - \mathbf{b} \) is \( 90^\circ \).

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