If `|vec(a)|=10 |vec(b)|=2 and vec(a).vec(b)=12`, then what is the value of `|vec(a) xx vec(b)|` ?

Options

A. 12
B. 16
C. 20
D. 24

Answer

To find the value of \(|\vec{a} \times \vec{b}|\), we can use the formula that relates the magnitudes of the vectors and their dot product:

\[ |\vec{a} \times \vec{b}| = \sqrt{|\vec{a}|^2 |\vec{b}|^2 - (\vec{a} \cdot \vec{b})^2} \]

### Step-by-Step Solution:

1. **Identify the given values:**
- \(|\vec{a}| = 10\)
- \(|\vec{b}| = 2\)
- \(\vec{a} \cdot \vec{b} = 12\)

2. **Substitute the values into the formula:**
\[ |\vec{a} \times \vec{b}| = \sqrt{|\vec{a}|^2 |\vec{b}|^2 - (\vec{a} \cdot \vec{b})^2} \]
becomes
\[ |\vec{a} \times \vec{b}| = \sqrt{(10^2)(2^2) - (12)^2} \]

3. **Calculate \(|\vec{a}|^2\) and \(|\vec{b}|^2\):**
- \(|\vec{a}|^2 = 10^2 = 100\)
- \(|\vec{b}|^2 = 2^2 = 4\)

4. **Calculate \((\vec{a} \cdot \vec{b})^2\):**
- \((\vec{a} \cdot \vec{b})^2 = 12^2 = 144\)

5. **Substitute these values back into the equation:**
\[ |\vec{a} \times \vec{b}| = \sqrt{(100)(4) - 144} \]

6. **Calculate \(100 \times 4\):**
- \(100 \times 4 = 400\)

7. **Now substitute back into the equation:**
\[ |\vec{a} \times \vec{b}| = \sqrt{400 - 144} \]

8. **Calculate \(400 - 144\):**
- \(400 - 144 = 256\)

9. **Finally, take the square root:**
\[ |\vec{a} \times \vec{b}| = \sqrt{256} = 16 \]

### Final Answer:
\[ |\vec{a} \times \vec{b}| = 16 \]