If ` veca=2 hat i+2 hat j+3 hat k , vecb=- hat i+2 hat j+ hat k` and ` vecc=3 hat i+ hat j` are such that ` veca+lambda vecb`is perpendicular to ` vecc`, then find the value of `lambda`.

Explanation

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

Given:
- \( \vec{a} = 2 \hat{i} + 2 \hat{j} + 3 \hat{k} \)
- \( \vec{b} = -\hat{i} + 2 \hat{j} + \hat{k} \)
- \( \vec{c} = 3 \hat{i} + \hat{j} \)

### Step 1: Write the expression for \( \vec{a} + \lambda \vec{b} \)
...