Examine whether the vectors `a=2hat(i)+3hat(j)+2hat(k), b=hat(i)-hat(j)+2hat(k) and c=4hat(i)+2hat(j)+4hat(k)` form a left handed or a right handed system.

Answer

To determine whether the vectors \( \mathbf{a} = 2\hat{i} + 3\hat{j} + 2\hat{k} \), \( \mathbf{b} = \hat{i} - \hat{j} + 2\hat{k} \), and \( \mathbf{c} = 4\hat{i} + 2\hat{j} + 4\hat{k} \) form a left-handed or right-handed system, we will calculate the box product (scalar triple product) of these vectors.

### Step-by-step Solution:

1. **Identify the vectors**:
- \( \mathbf{a} = 2\hat{i} + 3\hat{j} + 2\hat{k} \)
- \( \mathbf{b} = \hat{i} - \hat{j} + 2\hat{k} \)
- \( \mathbf{c} = 4\hat{i} + 2\hat{j} + 4\hat{k} \)

2. **Write the components of the vectors**:
- For \( \mathbf{a} \), the components are \( a_1 = 2, a_2 = 3, a_3 = 2 \).
- For \( \mathbf{b} \), the components are \( b_1 = 1, b_2 = -1, b_3 = 2 \).
- For \( \mathbf{c} \), the components are \( c_1 = 4, c_2 = 2, c_3 = 4 \).

3. **Set up the determinant for the box product**:
\[ \text{Box Product} = \begin{vmatrix} 2 & 3 & 2 \\ 1 & -1 & 2 \\ 4 & 2 & 4 \end{vmatrix} \]

4. **Calculate the determinant**:
- Using the first row to expand the determinant:
\[ = 2 \begin{vmatrix} -1 & 2 \\ 2 & 4 \end{vmatrix} - 3 \begin{vmatrix} 1 & 2 \\ 4 & 4 \end{vmatrix} + 2 \begin{vmatrix} 1 & -1 \\ 4 & 2 \end{vmatrix} \]

5. **Calculate the 2x2 determinants**:
- For the first determinant:
\[ \begin{vmatrix} -1 & 2 \\ 2 & 4 \end{vmatrix} = (-1)(4) - (2)(2) = -4 - 4 = -8 \]
- For the second determinant:
\[ \begin{vmatrix} 1 & 2 \\ 4 & 4 \end{vmatrix} = (1)(4) - (2)(4) = 4 - 8 = -4 \]
- For the third determinant:
\[ \begin{vmatrix} 1 & -1 \\ 4 & 2 \end{vmatrix} = (1)(2) - (-1)(4) = 2 + 4 = 6 \]

6. **Substituting back into the determinant**:
\[ = 2(-8) - 3(-4) + 2(6) \]
\[ = -16 + 12 + 12 \]
\[ = -16 + 24 = 8 \]

7. **Conclusion**:
- The box product is \( 8 \), which is positive. Therefore, the vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) form a right-handed system.