For what value of `lamda` are the vector `veca and vecb` perpendicular to each other ? Where `veca = lamda hati + 2hatj + hatk and vecb = 5 hati-9 hatj + 2hatk`

Answer

To find the value of \( \lambda \) for which the vectors \( \vec{a} \) and \( \vec{b} \) are perpendicular, we need to use the property that two vectors are perpendicular if their dot product is equal to zero.

Given:
\[ \vec{a} = \lambda \hat{i} + 2 \hat{j} + \hat{k} \]
\[ \vec{b} = 5 \hat{i} - 9 \hat{j} + 2 \hat{k} \]

### Step 1: Calculate the dot product of \( \vec{a} \) and \( \vec{b} \)
The dot product \( \vec{a} \cdot \vec{b} \) is calculated as follows:
\[ \vec{a} \cdot \vec{b} = (\lambda \hat{i} + 2 \hat{j} + \hat{k}) \cdot (5 \hat{i} - 9 \hat{j} + 2 \hat{k}) \]

### Step 2: Apply the formula for dot product
Using the dot product formula:
\[ \vec{a} \cdot \vec{b} = \lambda \cdot 5 + 2 \cdot (-9) + 1 \cdot 2 \]

### Step 3: Simplify the expression
Calculating each term:
\[ = 5\lambda - 18 + 2 \]
\[ = 5\lambda - 16 \]

### Step 4: Set the dot product to zero
Since the vectors are perpendicular, we set the dot product equal to zero:
\[ 5\lambda - 16 = 0 \]

### Step 5: Solve for \( \lambda \)
Rearranging the equation gives:
\[ 5\lambda = 16 \]
\[ \lambda = \frac{16}{5} \]

### Final Answer
The value of \( \lambda \) for which the vectors \( \vec{a} \) and \( \vec{b} \) are perpendicular is:
\[ \lambda = \frac{16}{5} \]