If `| vec(a) xx vec(b) | =4` and `| vec(a). vec(b) |=2`, then `| vec( a) |^(2) | vec( b) | ^(2)` is equal to

Options

A. 6
B. 8
C. 20
D. none of these

Answer

To solve the problem, we will use the relationship between the dot product and the cross product of two vectors.

Given:
1. \(|\vec{a} \times \vec{b}| = 4\)
2. \(|\vec{a} \cdot \vec{b}| = 2\)

We need to find \(|\vec{a}|^2 |\vec{b}|^2\).

### Step-by-Step Solution:

**Step 1: Use the relationship between dot product and cross product.**

We know the following relationship:
\[ |\vec{a} \times \vec{b}|^2 + |\vec{a} \cdot \vec{b}|^2 = |\vec{a}|^2 |\vec{b}|^2 \]

**Step 2: Substitute the given values into the equation.**

Substituting the values we have:
\[ |\vec{a} \times \vec{b}|^2 = 4^2 = 16 \]
\[ |\vec{a} \cdot \vec{b}|^2 = 2^2 = 4 \]

**Step 3: Plug these values into the relationship.**

Now substituting these into the equation:
\[ 16 + 4 = |\vec{a}|^2 |\vec{b}|^2 \]

**Step 4: Simplify the equation.**

Adding the left-hand side:
\[ 20 = |\vec{a}|^2 |\vec{b}|^2 \]

Thus, we find:
\[ |\vec{a}|^2 |\vec{b}|^2 = 20 \]

### Final Answer:
\[ |\vec{a}|^2 |\vec{b}|^2 = 20 \]