If `a=hati+2hatj+3hatk,b=-hati+2hatj+hatk and c=3hati+hatj`. If `(a+tb) bot c`, then t is equal to

Options

A. 5
B. 4
C. 3
D. 2

Answer

To solve the problem step by step, we will follow the reasoning outlined in the video transcript:

### Step 1: Write down the vectors
Given:
- \( \mathbf{a} = \hat{i} + 2\hat{j} + 3\hat{k} \)
- \( \mathbf{b} = -\hat{i} + 2\hat{j} + \hat{k} \)
- \( \mathbf{c} = 3\hat{i} + \hat{j} \)

### Step 2: Set up the condition for perpendicularity
Since \( \mathbf{a} + t\mathbf{b} \) is perpendicular to \( \mathbf{c} \), we know that their dot product must equal zero:
\[ (\mathbf{a} + t\mathbf{b}) \cdot \mathbf{c} = 0 \]

### Step 3: Express \( \mathbf{a} + t\mathbf{b} \)
First, we calculate \( t\mathbf{b} \):
\[ t\mathbf{b} = t(-\hat{i} + 2\hat{j} + \hat{k}) = -t\hat{i} + 2t\hat{j} + t\hat{k} \]

Now, we can add \( \mathbf{a} \) and \( t\mathbf{b} \):
\[ \mathbf{a} + t\mathbf{b} = (\hat{i} + 2\hat{j} + 3\hat{k}) + (-t\hat{i} + 2t\hat{j} + t\hat{k}) \]
\[ = (1 - t)\hat{i} + (2 + 2t)\hat{j} + (3 + t)\hat{k} \]

### Step 4: Calculate the dot product with \( \mathbf{c} \)
Now we will compute the dot product:
\[ ((1 - t)\hat{i} + (2 + 2t)\hat{j} + (3 + t)\hat{k}) \cdot (3\hat{i} + \hat{j}) \]
This expands to:
\[ (1 - t) \cdot 3 + (2 + 2t) \cdot 1 + (3 + t) \cdot 0 = 0 \]
\[ 3(1 - t) + (2 + 2t) = 0 \]

### Step 5: Simplify the equation
Expanding the equation gives:
\[ 3 - 3t + 2 + 2t = 0 \]
Combining like terms:
\[ 5 - t = 0 \]

### Step 6: Solve for \( t \)
Rearranging gives:
\[ t = 5 \]

### Conclusion
The value of \( t \) is \( 5 \).