The scalar product of the vector ` hat i+ hat j+ hat k`with a unit vector along the sum of vector `2 hat i+4 hat j-5 hat k`and `lambda hat i+2 hat j+3 hat k`is equal to one. Find the value of `lambda`.

Explanation

To solve the problem, we need to find the value of \( \lambda \) such that the scalar product of the vector \( \hat{i} + \hat{j} + \hat{k} \) with a unit vector along the sum of the vectors \( 2\hat{i} + 4\hat{j} - 5\hat{k} \) and \( \lambda \hat{i} + 2\hat{j} + 3\hat{k} \) equals 1.

### Step-by-Step Solution:

1. **Define the vectors**:
Let \( \mathbf{a} = \hat{i} + \hat{j} + \hat{k} \) and \( \mathbf{b} = (2\hat{i} + 4\hat{j} - 5\hat{k}) + (\lambda \hat{i} + 2\hat{j} + 3\hat{k}) \).

2. **Combine the vectors**:
...