The vector `vecB= 5hati+2hatj-Shatk` is perpendicular to the vector `vecA= 3hati+hatj+2hatk` if S =

Options

A. 1
B. 4.7
C. 6.3
D. 8.5

Explanation

To determine the value of \( S \) for which the vector \( \vec{B} = 5\hat{i} + 2\hat{j} - S\hat{k} \) is perpendicular to the vector \( \vec{A} = 3\hat{i} + \hat{j} + 2\hat{k} \), we can use the property that two vectors are perpendicular if their dot product is zero.

### Step-by-Step Solution:

1. **Write down the vectors**:
\[ \vec{A} = 3\hat{i} + \hat{j} + 2\hat{k} \]
\[ \vec{B} = 5\hat{i} + 2\hat{j} - S\hat{k} \]

2. **Set up the dot product**:
The dot product \( \vec{A} \cdot \vec{B} \) is given by:
\[ \vec{A} \cdot \vec{B} = (3\hat{i} + \hat{j} + 2\hat{k}) \cdot (5\hat{i} + 2\hat{j} - S\hat{k}) \]

3. **Calculate the dot product**:
Using the properties of dot products:
\[ \vec{A} \cdot \vec{B} = 3 \cdot 5 + 1 \cdot 2 + 2 \cdot (-S) \]
Simplifying this gives:
\[ = 15 + 2 - 2S \]
Therefore:
\[ \vec{A} \cdot \vec{B} = 17 - 2S \]

4. **Set the dot product to zero** (since the vectors are perpendicular):
\[ 17 - 2S = 0 \]

5. **Solve for \( S \)**:
Rearranging the equation:
\[ 2S = 17 \]
Dividing both sides by 2:
\[ S = \frac{17}{2} = 8.5 \]

### Final Answer:
The value of \( S \) for which vector \( \vec{B} \) is perpendicular to vector \( \vec{A} \) is:
\[ S = 8.5 \]