Find a vector of magnittude` sqrt(171)` which is perpendicular to both of the vectors ` hata = hati + 2 hatj - 3 hatk and hatb = 3 hati - hatj + 2hatk.`

Answer

To find a vector of magnitude \( \sqrt{171} \) that is perpendicular to both vectors \( \hat{a} = \hat{i} + 2\hat{j} - 3\hat{k} \) and \( \hat{b} = 3\hat{i} - \hat{j} + 2\hat{k} \), we can follow these steps:

### Step 1: Write down the vectors
Given:
\[ \hat{a} = \hat{i} + 2\hat{j} - 3\hat{k} \]
\[ \hat{b} = 3\hat{i} - \hat{j} + 2\hat{k} \]

### Step 2: Find the cross product \( \hat{a} \times \hat{b} \)
The cross product of two vectors \( \hat{a} \) and \( \hat{b} \) can be computed using the determinant of a matrix formed by the unit vectors and the components of the vectors.

\[ \hat{c} = \hat{a} \times \hat{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & -3 \\ 3 & -1 & 2 \end{vmatrix} \]

Calculating the determinant:
\[ \hat{c} = \hat{i} \begin{vmatrix} 2 & -3 \\ -1 & 2 \end{vmatrix} - \hat{j} \begin{vmatrix} 1 & -3 \\ 3 & 2 \end{vmatrix} + \hat{k} \begin{vmatrix} 1 & 2 \\ 3 & -1 \end{vmatrix} \]

Calculating each of the 2x2 determinants:
1. \( \begin{vmatrix} 2 & -3 \\ -1 & 2 \end{vmatrix} = (2)(2) - (-3)(-1) = 4 - 3 = 1 \)
2. \( \begin{vmatrix} 1 & -3 \\ 3 & 2 \end{vmatrix} = (1)(2) - (-3)(3) = 2 + 9 = 11 \)
3. \( \begin{vmatrix} 1 & 2 \\ 3 & -1 \end{vmatrix} = (1)(-1) - (2)(3) = -1 - 6 = -7 \)

Putting it all together:
\[ \hat{c} = \hat{i}(1) - \hat{j}(11) + \hat{k}(-7) = \hat{i} - 11\hat{j} - 7\hat{k} \]

### Step 3: Find the magnitude of \( \hat{c} \)
The magnitude of \( \hat{c} \) is calculated as follows:
\[ |\hat{c}| = \sqrt{(1)^2 + (-11)^2 + (-7)^2} = \sqrt{1 + 121 + 49} = \sqrt{171} \]

### Step 4: Conclusion
Since \( \hat{c} \) is perpendicular to both \( \hat{a} \) and \( \hat{b} \) and has the required magnitude \( \sqrt{171} \), we can conclude:
\[ \hat{c} = \hat{i} - 11\hat{j} - 7\hat{k} \]

### Final Answer
The vector of magnitude \( \sqrt{171} \) that is perpendicular to both vectors is:
\[ \hat{c} = \hat{i} - 11\hat{j} - 7\hat{k} \]

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