If `vec(a)=vec(i)+2 hat(j)-3hat(k) and vec(b)=3hat(i)-hat(j)+lambda hat(k) and (vec(a)+vec(b))` is perpendicular to `vec(a)-vec(b)`, then what is the value of `lambda`?

Options

A. `-2` only
B. `pm2`
C. 3 only
D. `pm3`

Explanation

To solve the problem, we need to find the value of \( \lambda \) given that \( \vec{a} + \vec{b} \) is perpendicular to \( \vec{a} - \vec{b} \).

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

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

### Step 3: Calculate \( \vec{a} - \vec{b} \)
\[ \vec{a} - \vec{b} = (\hat{i} + 2\hat{j} - 3\hat{k}) - (3\hat{i} - \hat{j} + \lambda \hat{k}) \]
Combine the components:
\[ = (1 - 3)\hat{i} + (2 + 1)\hat{j} + (-3 - \lambda)\hat{k} \]
\[ = -2\hat{i} + 3\hat{j} + (-3 - \lambda)\hat{k} \]

### Step 4: Set up the dot product condition
Since \( \vec{a} + \vec{b} \) is perpendicular to \( \vec{a} - \vec{b} \), we have:
\[ (\vec{a} + \vec{b}) \cdot (\vec{a} - \vec{b}) = 0 \]
Substituting the vectors:
\[ (4\hat{i} + 1\hat{j} + (\lambda - 3)\hat{k}) \cdot (-2\hat{i} + 3\hat{j} + (-3 - \lambda)\hat{k}) = 0 \]

### Step 5: Calculate the dot product
Calculating the dot product:
\[ = 4 \cdot (-2) + 1 \cdot 3 + (\lambda - 3)(-3 - \lambda) \]
\[ = -8 + 3 + (-3\lambda + 9 + \lambda^2 - 3\lambda) \]
\[ = -5 + \lambda^2 - 6\lambda + 9 = 0 \]
\[ \lambda^2 - 6\lambda + 4 = 0 \]

### Step 6: Solve the quadratic equation
Using the quadratic formula:
\[ \lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -6, c = 4 \):
\[ \lambda = \frac{6 \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot 4}}{2 \cdot 1} \]
\[ = \frac{6 \pm \sqrt{36 - 16}}{2} \]
\[ = \frac{6 \pm \sqrt{20}}{2} \]
\[ = \frac{6 \pm 2\sqrt{5}}{2} \]
\[ = 3 \pm \sqrt{5} \]

### Final Answer
Thus, the values of \( \lambda \) are:
\[ \lambda = 3 + \sqrt{5} \quad \text{or} \quad \lambda = 3 - \sqrt{5} \]