Let `bara = bari + 2barj+ 3bark , barb = -bari + 2barj+bark. " If " barc = 3bari +barj+tbark` is perpendicular to `(bara+barb)`, then find t.

Answer

Let's solve the problem step by step.

### Step 1: Define the vectors
We are given the following vectors:
- \( \vec{a} = \hat{i} + 2\hat{j} + 3\hat{k} \)
- \( \vec{b} = -\hat{i} + 2\hat{j} + \hat{k} \)
- \( \vec{c} = 3\hat{i} + \hat{j} + t\hat{k} \)

### Step 2: Calculate \( \vec{a} + \vec{b} \)
Now, we need to find the sum of vectors \( \vec{a} \) and \( \vec{b} \):
\[ \vec{a} + \vec{b} = (\hat{i} + 2\hat{j} + 3\hat{k}) + (-\hat{i} + 2\hat{j} + \hat{k}) \]
Combining like terms:
\[ = (1 - 1)\hat{i} + (2 + 2)\hat{j} + (3 + 1)\hat{k} = 0\hat{i} + 4\hat{j} + 4\hat{k} \]
Thus,
\[ \vec{a} + \vec{b} = 4\hat{j} + 4\hat{k} \]

### Step 3: Use the condition of perpendicularity
Since \( \vec{c} \) is perpendicular to \( \vec{a} + \vec{b} \), we can use the dot product:
\[ (\vec{a} + \vec{b}) \cdot \vec{c} = 0 \]
Substituting the vectors:
\[ (4\hat{j} + 4\hat{k}) \cdot (3\hat{i} + \hat{j} + t\hat{k}) = 0 \]

### Step 4: Calculate the dot product
Now, we calculate the dot product:
\[ = 4\hat{j} \cdot 3\hat{i} + 4\hat{j} \cdot \hat{j} + 4\hat{k} \cdot (t\hat{k}) \]
Since \( \hat{i} \cdot \hat{j} = 0 \) and \( \hat{j} \cdot \hat{j} = 1 \) and \( \hat{k} \cdot \hat{k} = 1 \):
\[ = 0 + 4 \cdot 1 + 4t \cdot 1 = 4 + 4t \]

### Step 5: Set the dot product equal to zero
Setting the result equal to zero:
\[ 4 + 4t = 0 \]

### Step 6: Solve for \( t \)
Now, we solve for \( t \):
\[ 4t = -4 \implies t = -1 \]

### Final Answer
Thus, the value of \( t \) is:
\[ \boxed{-1} \]