Find a vector of magnitude 6 which is perpendicular to both the vectors `2 hati - hatj + 2hatk and 4 hati - hatj + 3hatk.`

Answer

To find a vector of magnitude 6 that is perpendicular to both given vectors \( \mathbf{A} = 2 \hat{i} - \hat{j} + 2 \hat{k} \) and \( \mathbf{B} = 4 \hat{i} - \hat{j} + 3 \hat{k} \), we can follow these steps:

### Step 1: Calculate the Cross Product \( \mathbf{A} \times \mathbf{B} \)

The cross product of two vectors \( \mathbf{A} \) and \( \mathbf{B} \) is given by the determinant of a matrix formed by the unit vectors \( \hat{i}, \hat{j}, \hat{k} \) and the components of the vectors.

\[ \mathbf{A} \times \mathbf{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & -1 & 2 \\ 4 & -1 & 3 \end{vmatrix} \]

Calculating the determinant:

\[ = \hat{i} \begin{vmatrix} -1 & 2 \\ -1 & 3 \end{vmatrix} - \hat{j} \begin{vmatrix} 2 & 2 \\ 4 & 3 \end{vmatrix} + \hat{k} \begin{vmatrix} 2 & -1 \\ 4 & -1 \end{vmatrix} \]

Calculating each of these determinants:

1. For \( \hat{i} \):
\[ (-1)(3) - (-1)(2) = -3 + 2 = -1 \]

2. For \( \hat{j} \):
\[ (2)(3) - (2)(4) = 6 - 8 = -2 \]

3. For \( \hat{k} \):
\[ (2)(-1) - (-1)(4) = -2 + 4 = 2 \]

Putting this together, we have:
\[ \mathbf{A} \times \mathbf{B} = -1 \hat{i} + 2 \hat{j} + 2 \hat{k} \]

### Step 2: Find the Magnitude of \( \mathbf{A} \times \mathbf{B} \)

The magnitude of the vector \( \mathbf{A} \times \mathbf{B} \) is calculated as follows:

\[ |\mathbf{A} \times \mathbf{B}| = \sqrt{(-1)^2 + 2^2 + 2^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 \]

### Step 3: Find the Unit Vector of \( \mathbf{A} \times \mathbf{B} \)

To find the unit vector, we divide each component of the cross product by its magnitude:

\[ \text{Unit vector} = \frac{\mathbf{A} \times \mathbf{B}}{|\mathbf{A} \times \mathbf{B}|} = \frac{-1 \hat{i} + 2 \hat{j} + 2 \hat{k}}{3} = -\frac{1}{3} \hat{i} + \frac{2}{3} \hat{j} + \frac{2}{3} \hat{k} \]

### Step 4: Scale the Unit Vector to Magnitude 6

To find a vector of magnitude 6, we multiply the unit vector by 6:

\[ \mathbf{R} = 6 \left(-\frac{1}{3} \hat{i} + \frac{2}{3} \hat{j} + \frac{2}{3} \hat{k}\right) = -2 \hat{i} + 4 \hat{j} + 4 \hat{k} \]

### Step 5: Verify the Magnitude of \( \mathbf{R} \)

Finally, we can check the magnitude of \( \mathbf{R} \):

\[ |\mathbf{R}| = \sqrt{(-2)^2 + 4^2 + 4^2} = \sqrt{4 + 16 + 16} = \sqrt{36} = 6 \]

Thus, the required vector is:

\[ \mathbf{R} = -2 \hat{i} + 4 \hat{j} + 4 \hat{k} \]