Consider the vectors `a = i- 2j + k and b= 4i- 4j + 7k`
What is the vector perpendicular to both the vectors?

Options

A. `-10i -3j +4k`
B. `-10i + 3j+4k`
C. `10i-3j + 4k`
D. None of these

Answer

To find a vector that is perpendicular to both vectors \( \mathbf{a} \) and \( \mathbf{b} \), we can use the cross product. The cross product of two vectors results in a vector that is orthogonal to both of the original vectors.

### Step-by-Step Solution:

1. **Identify the vectors:**
Given vectors are:
\[ \mathbf{a} = \mathbf{i} - 2\mathbf{j} + \mathbf{k} \]
\[ \mathbf{b} = 4\mathbf{i} - 4\mathbf{j} + 7\mathbf{k} \]

2. **Set up the cross product:**
The cross product \( \mathbf{a} \times \mathbf{b} \) can be calculated using the determinant of a matrix formed by the unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) and the components of the vectors \( \mathbf{a} \) and \( \mathbf{b} \):
\[ \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & -2 & 1 \\ 4 & -4 & 7 \end{vmatrix} \]

3. **Calculate the determinant:**
We can expand the determinant as follows:
\[ \mathbf{a} \times \mathbf{b} = \mathbf{i} \begin{vmatrix} -2 & 1 \\ -4 & 7 \end{vmatrix} - \mathbf{j} \begin{vmatrix} 1 & 1 \\ 4 & 7 \end{vmatrix} + \mathbf{k} \begin{vmatrix} 1 & -2 \\ 4 & -4 \end{vmatrix} \]

Now, we calculate each of the 2x2 determinants:
- For \( \mathbf{i} \):
\[ \begin{vmatrix} -2 & 1 \\ -4 & 7 \end{vmatrix} = (-2)(7) - (1)(-4) = -14 + 4 = -10 \]
- For \( \mathbf{j} \):
\[ \begin{vmatrix} 1 & 1 \\ 4 & 7 \end{vmatrix} = (1)(7) - (1)(4) = 7 - 4 = 3 \]
- For \( \mathbf{k} \):
\[ \begin{vmatrix} 1 & -2 \\ 4 & -4 \end{vmatrix} = (1)(-4) - (-2)(4) = -4 + 8 = 4 \]

4. **Combine the results:**
Now substituting back into the equation for \( \mathbf{a} \times \mathbf{b} \):
\[ \mathbf{a} \times \mathbf{b} = -10\mathbf{i} - 3\mathbf{j} + 4\mathbf{k} \]

5. **Final result:**
Therefore, the vector that is perpendicular to both \( \mathbf{a} \) and \( \mathbf{b} \) is:
\[ \mathbf{a} \times \mathbf{b} = -10\mathbf{i} - 3\mathbf{j} + 4\mathbf{k} \]