The projection of `hati+hatj+hatk` on the whole equation is `vecr=(3+lamda)hati+(2lamda-1)hatj+3lamda hatk, lamda` being the scalar parameter is:

Options

A. `3/(sqrt(11))`
B. `6/(sqrt(14))`
C. `6/(sqrt(11))`
D. `3/(sqrt(14))`

Explanation

To solve the problem, we need to find the scalar parameter \( \lambda \) such that the projection of the vector \( \hat{i} + \hat{j} + \hat{k} \) on the direction vector given by the equation \( \vec{r} = (3 + \lambda) \hat{i} + (2\lambda - 1) \hat{j} + 3\lambda \hat{k} \) is equal to a specific value.

### Step-by-Step Solution:

1. **Identify the vectors**:
- Let \( \vec{c} = \hat{i} + \hat{j} + \hat{k} \).
- The direction vector \( \vec{b} \) can be extracted from the equation \( \vec{r} = (3 + \lambda) \hat{i} + (2\lambda - 1) \hat{j} + 3\lambda \hat{k} \). Thus, \( \vec{b} = \hat{i} + (2\lambda - 1) \hat{j} + 3\lambda \hat{k} \).

...