Step 1: Understanding the Concept:
The cross product of two vectors is anti-commutative. This means that the direction of the product vector flips when the order of the operands is reversed. Step 2: Detailed Explanation:
By the property of vector products (cross products):
\[ \vec{a} \times \vec{b} = - (\vec{b} \times \vec{a}) \]
Alternatively:
\[ \vec{b} \times \vec{a} = - (\vec{a} \times \vec{b}) \]
Now, look at the expression inside the magnitude sign:
\[ \vec{a} \times \vec{b} + \vec{b} \times \vec{a} \]
Substituting \( \vec{b} \times \vec{a} = - (\vec{a} \times \vec{b}) \):
\[ = \vec{a} \times \vec{b} + (- \vec{a} \times \vec{b}) \]
\[ = \vec{a} \times \vec{b} - \vec{a} \times \vec{b} = \vec{0} \]
The magnitude of a zero vector is 0.
\[ |\vec{0}| = 0 \]. Step 3: Final Answer:
The value is 0.