If the vectors `2 hati +3 hat j +chatj` and `-3hat i+6hatk` are orthogonal, the value of c is :

Options

A. 0
B. `-1`
C. 1
D. `1//2`

Explanation

To find the value of \( c \) such that the vectors \( \mathbf{A} = 2 \hat{i} + 3 \hat{j} + c \hat{k} \) and \( \mathbf{B} = -3 \hat{i} + 6 \hat{k} \) are orthogonal, we will follow these steps:

### Step 1: Understand the condition for orthogonality
Two vectors are orthogonal if their dot product is zero. Therefore, we need to compute the dot product \( \mathbf{A} \cdot \mathbf{B} \) and set it equal to zero.

### Step 2: Write down the dot product
The dot product of two vectors \( \mathbf{A} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \) and \( \mathbf{B} = b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} \) is given by:
\[ \mathbf{A} \cdot \mathbf{B} = a_1 b_1 + a_2 b_2 + a_3 b_3 \]
For our vectors:
- \( a_1 = 2 \), \( a_2 = 3 \), \( a_3 = c \)
- \( b_1 = -3 \), \( b_2 = 0 \), \( b_3 = 6 \)

### Step 3: Substitute the components into the dot product formula
Now we can substitute the values into the dot product:
\[ \mathbf{A} \cdot \mathbf{B} = (2)(-3) + (3)(0) + (c)(6) \]

### Step 4: Simplify the expression
Calculating the terms gives:
\[ \mathbf{A} \cdot \mathbf{B} = -6 + 0 + 6c = -6 + 6c \]

### Step 5: Set the dot product equal to zero
Since the vectors are orthogonal, we set the dot product to zero:
\[ -6 + 6c = 0 \]

### Step 6: Solve for \( c \)
Now, we can solve for \( c \):
\[ 6c = 6 \\ c = 1 \]

### Conclusion
Thus, the value of \( c \) is \( 1 \).