If `a = i + 2j - 3k, b = 3i - j + 2k`, then angle between a + b and a - b is :

Options

A. 0
B. `30^(@)`
C. `60^(@)`
D. `90^(@)`

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:
\[ \mathbf{a} = \mathbf{i} + 2\mathbf{j} - 3\mathbf{k} \]
\[ \mathbf{b} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k} \]

### Step 2: Calculate \( \mathbf{a} + \mathbf{b} \)
To find \( \mathbf{a} + \mathbf{b} \):
\[ \mathbf{a} + \mathbf{b} = (\mathbf{i} + 2\mathbf{j} - 3\mathbf{k}) + (3\mathbf{i} - \mathbf{j} + 2\mathbf{k}) \]
Combine the components:
\[ = (1 + 3)\mathbf{i} + (2 - 1)\mathbf{j} + (-3 + 2)\mathbf{k} \]
\[ = 4\mathbf{i} + \mathbf{j} - \mathbf{k} \]

### Step 3: Calculate \( \mathbf{a} - \mathbf{b} \)
To find \( \mathbf{a} - \mathbf{b} \):
\[ \mathbf{a} - \mathbf{b} = (\mathbf{i} + 2\mathbf{j} - 3\mathbf{k}) - (3\mathbf{i} - \mathbf{j} + 2\mathbf{k}) \]
Combine the components:
\[ = (1 - 3)\mathbf{i} + (2 + 1)\mathbf{j} + (-3 - 2)\mathbf{k} \]
\[ = -2\mathbf{i} + 3\mathbf{j} - 5\mathbf{k} \]

### Step 4: Find the dot product \( (\mathbf{a} + \mathbf{b}) \cdot (\mathbf{a} - \mathbf{b}) \)
Now calculate the dot product:
\[ (4\mathbf{i} + \mathbf{j} - \mathbf{k}) \cdot (-2\mathbf{i} + 3\mathbf{j} - 5\mathbf{k}) \]
Calculating each term:
\[ = (4)(-2) + (1)(3) + (-1)(-5) \]
\[ = -8 + 3 + 5 = 0 \]

### Step 5: Use the dot product to find the angle
The dot product formula gives:
\[ \mathbf{u} \cdot \mathbf{v} = |\mathbf{u}| |\mathbf{v}| \cos \theta \]
Since the dot product is zero:
\[ 0 = |\mathbf{u}| |\mathbf{v}| \cos \theta \]
This implies:
\[ \cos \theta = 0 \]
Thus, the angle \( \theta \) is:
\[ \theta = \frac{\pi}{2} \text{ radians} \quad \text{or} \quad 90^\circ \]

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

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