If `vec(a)=5hat(i)-hat(j)+7hat(k)` and `vec(b)=hat(i)-hat(j)-lambda hat(k)`, find the value of `lambda` for which `(vec(a)+vec(b))` and `(vec(a)-vec(b))` are orthogonal.

Answer

To solve the problem, we need to find the value of \( \lambda \) such that the vectors \( \vec{a} + \vec{b} \) and \( \vec{a} - \vec{b} \) are orthogonal. This means that their dot product should equal zero.

### Step-by-Step Solution:

1. **Define the Vectors**:
\[ \vec{a} = 5\hat{i} - \hat{j} + 7\hat{k} \]
\[ \vec{b} = \hat{i} - \hat{j} - \lambda \hat{k} \]

2. **Calculate \( \vec{a} + \vec{b} \)**:
\[ \vec{a} + \vec{b} = (5\hat{i} - \hat{j} + 7\hat{k}) + (\hat{i} - \hat{j} - \lambda \hat{k}) \]
Combine the components:
\[ \vec{a} + \vec{b} = (5 + 1)\hat{i} + (-1 - 1)\hat{j} + (7 - \lambda)\hat{k} \]
\[ \vec{a} + \vec{b} = 6\hat{i} - 2\hat{j} + (7 - \lambda)\hat{k} \]

3. **Calculate \( \vec{a} - \vec{b} \)**:
\[ \vec{a} - \vec{b} = (5\hat{i} - \hat{j} + 7\hat{k}) - (\hat{i} - \hat{j} - \lambda \hat{k}) \]
Combine the components:
\[ \vec{a} - \vec{b} = (5 - 1)\hat{i} + (-1 + 1)\hat{j} + (7 + \lambda)\hat{k} \]
\[ \vec{a} - \vec{b} = 4\hat{i} + 0\hat{j} + (7 + \lambda)\hat{k} \]

4. **Set Up the Dot Product**:
The vectors \( \vec{a} + \vec{b} \) and \( \vec{a} - \vec{b} \) are orthogonal if:
\[ (\vec{a} + \vec{b}) \cdot (\vec{a} - \vec{b}) = 0 \]

5. **Calculate the Dot Product**:
\[ (6\hat{i} - 2\hat{j} + (7 - \lambda)\hat{k}) \cdot (4\hat{i} + 0\hat{j} + (7 + \lambda)\hat{k}) \]
Calculate each component:
\[ = 6 \cdot 4 + (-2) \cdot 0 + (7 - \lambda)(7 + \lambda) \]
\[ = 24 + (7^2 - \lambda^2) = 24 + 49 - \lambda^2 \]
\[ = 73 - \lambda^2 \]

6. **Set the Dot Product to Zero**:
\[ 73 - \lambda^2 = 0 \]
\[ \lambda^2 = 73 \]

7. **Solve for \( \lambda \)**:
\[ \lambda = \pm \sqrt{73} \]

### Final Answer:
The values of \( \lambda \) for which \( \vec{a} + \vec{b} \) and \( \vec{a} - \vec{b} \) are orthogonal are:
\[ \lambda = \sqrt{73} \quad \text{or} \quad \lambda = -\sqrt{73} \]