Find the value of `lamda` such that the vectors `veca=2hati+lamdahatj+hatkandvecb=hati+2hatj+3hatk` are orthogonal.

Options

A. 0
B. 1
C. `(3)/(2)`
D. `(-5)/(2)`

Explanation

To find the value of \( \lambda \) such that the vectors \( \vec{a} = 2\hat{i} + \lambda\hat{j} + \hat{k} \) and \( \vec{b} = \hat{i} + 2\hat{j} + 3\hat{k} \) are orthogonal, we follow these steps:

### Step 1: Understand the Condition for Orthogonality
Two vectors are orthogonal if their dot product is zero. Therefore, we need to set up the equation:
\[ \vec{a} \cdot \vec{b} = 0 \]

### Step 2: Write the Dot Product
Calculate the dot product of \( \vec{a} \) and \( \vec{b} \):
\[ \vec{a} \cdot \vec{b} = (2\hat{i} + \lambda\hat{j} + \hat{k}) \cdot (\hat{i} + 2\hat{j} + 3\hat{k}) \]

### Step 3: Apply the Dot Product Formula
Using the properties of the dot product:
\[ \vec{a} \cdot \vec{b} = 2 \cdot 1 + \lambda \cdot 2 + 1 \cdot 3 \]
This simplifies to:
\[ \vec{a} \cdot \vec{b} = 2 + 2\lambda + 3 \]

### Step 4: Set the Dot Product to Zero
Set the equation equal to zero:
\[ 2 + 2\lambda + 3 = 0 \]
Combine the constants:
\[ 5 + 2\lambda = 0 \]

### Step 5: Solve for \( \lambda \)
Rearranging the equation gives:
\[ 2\lambda = -5 \]
Dividing both sides by 2:
\[ \lambda = -\frac{5}{2} \]

### Final Answer
The value of \( \lambda \) such that the vectors \( \vec{a} \) and \( \vec{b} \) are orthogonal is:
\[ \lambda = -\frac{5}{2} \]