Can you find at least one vector perpendiculr to `3hat(i)-4hat(j)+7hat(k)`?

Explanation

To find a vector that is perpendicular to the vector \( \mathbf{A} = 3\hat{i} - 4\hat{j} + 7\hat{k} \), we can use the concept of the dot product. Two vectors are perpendicular if their dot product is zero.

### Step-by-Step Solution:

1. **Define the Given Vector**:
\[ \mathbf{A} = 3\hat{i} - 4\hat{j} + 7\hat{k} \]

2. **Assume a General Vector**:
Let the vector we are looking for be \( \mathbf{B} = x\hat{i} + y\hat{j} + z\hat{k} \).

3. **Set Up the Dot Product Equation**:
For the vectors \( \mathbf{A} \) and \( \mathbf{B} \) to be perpendicular, we need:
\[ \mathbf{A} \cdot \mathbf{B} = 0 \]

This gives us:
\[ (3\hat{i} - 4\hat{j} + 7\hat{k}) \cdot (x\hat{i} + y\hat{j} + z\hat{k}) = 0 \]

4. **Calculate the Dot Product**:
The dot product can be calculated as:
\[ 3x - 4y + 7z = 0 \]

5. **Choose Values for \( x \) and \( y \)**:
To find a specific solution, we can choose arbitrary values for \( x \) and \( y \). For example, let’s choose \( x = 1 \) and \( y = 1 \).

6. **Solve for \( z \)**:
Substitute \( x = 1 \) and \( y = 1 \) into the equation:
\[ 3(1) - 4(1) + 7z = 0 \]
Simplifying gives:
\[ 3 - 4 + 7z = 0 \implies -1 + 7z = 0 \implies 7z = 1 \implies z = \frac{1}{7} \]

7. **Construct the Perpendicular Vector**:
Thus, one vector that is perpendicular to \( \mathbf{A} \) is:
\[ \mathbf{B} = 1\hat{i} + 1\hat{j} + \frac{1}{7}\hat{k} \]

### Final Answer:
A vector perpendicular to \( 3\hat{i} - 4\hat{j} + 7\hat{k} \) is:
\[ \mathbf{B} = \hat{i} + \hat{j} + \frac{1}{7}\hat{k} \]