Step 1: Understanding the Question:
The question requires calculating the cross product of two vectors and finding its magnitude.
The topic belongs to Vector Algebra.
Step 2: Key Formula or Approach:
The cross product \( \vec{a} \times \vec{b} \) is calculated using the determinant:
\[ \vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k}
a_1 & a_2 & a_3
b_1 & b_2 & b_3 \end{vmatrix} \]
The magnitude is given by \( |\vec{V}| = \sqrt{x^2 + y^2 + z^2} \).
Step 3: Detailed Explanation:
Given \( \vec{a} = (2, -1, 1) \) and \( \vec{b} = (1, 2, -3) \).
\[ \vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k}
2 & -1 & 1
1 & 2 & -3 \end{vmatrix} \]
Expanding along the first row:
\[ \vec{a} \times \vec{b} = \hat{i}[(-1)(-3) - (2)(1)] - \hat{j}[(2)(-3) - (1)(1)] + \hat{k}[(2)(2) - (1)(-1)] \]
\[ \vec{a} \times \vec{b} = \hat{i}(3 - 2) - \hat{j}(-6 - 1) + \hat{k}(4 + 1) \]
\[ \vec{a} \times \vec{b} = \hat{i} + 7\hat{j} + 5\hat{k} \]
Calculating the magnitude:
\[ |\vec{a} \times \vec{b}| = \sqrt{1^2 + 7^2 + 5^2} = \sqrt{1 + 49 + 25} = \sqrt{75} \]
Step 4: Final Answer:
Based on the provided options, the calculated magnitude is \( \sqrt{75} \).