If `|vec(a) xx vec(b)|^(2) + |vec(a) .vec(b)|^(2) = 400` and `|vec(a)| = 5`, then write the value of `|vec(b)|`.

Answer

To solve the problem, we start with the given equation:

\[ |\vec{a} \times \vec{b}|^2 + |\vec{a} \cdot \vec{b}|^2 = 400 \]

We also know that:

\[ |\vec{a}| = 5 \]

### Step 1: Use the identity for the magnitudes
We can use the identity that relates the magnitudes of the cross product and dot product:

\[ |\vec{a} \times \vec{b}|^2 + |\vec{a} \cdot \vec{b}|^2 = |\vec{a}|^2 |\vec{b}|^2 \]

### Step 2: Substitute the known values
Substituting the known value of \(|\vec{a}|\):

\[ |\vec{a}|^2 = 5^2 = 25 \]

So the equation becomes:

\[ |\vec{a}|^2 |\vec{b}|^2 = 400 \]

Substituting \(|\vec{a}|^2\):

\[ 25 |\vec{b}|^2 = 400 \]

### Step 3: Solve for \(|\vec{b}|^2\)
Now we can solve for \(|\vec{b}|^2\):

\[ |\vec{b}|^2 = \frac{400}{25} \]

Calculating the right side:

\[ |\vec{b}|^2 = 16 \]

### Step 4: Find \(|\vec{b}|\)
Now, we take the square root to find \(|\vec{b}|\):

\[ |\vec{b}| = \sqrt{16} = 4 \]

Thus, the value of \(|\vec{b}|\) is:

\[ |\vec{b}| = 4 \]

### Final Answer
\[ |\vec{b}| = 4 \]

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