Two vectors `vecA` and `vecB` have magnitude in the ratio `1:2` respectively. Their difference vector has as magnitude of 10 units, and the angle between `vecA & vecB` is `120^(@)` . The magnitude of `vecA and vecB` are :

Options

A. `(5)/(sqrt(7)) , (10)/(sqrt(7))`
B. `(5)/(sqrt(3)) , (10)/(sqrt(3))`
C. `(10)/(sqrt(7)) , (20)/(sqrt(7))`
D. `(10)/(sqrt(3)) , (20)/(sqrt(3))`

Explanation

To solve the problem, we need to find the magnitudes of vectors \( \vec{A} \) and \( \vec{B} \) given the following information:

1. The magnitudes of \( \vec{A} \) and \( \vec{B} \) are in the ratio \( 1:2 \).
2. The magnitude of the difference vector \( \vec{B} - \vec{A} \) is \( 10 \) units.
3. The angle between \( \vec{A} \) and \( \vec{B} \) is \( 120^\circ \).

### Step-by-Step Solution:

**Step 1: Define the magnitudes of the vectors.**
Let the magnitude of \( \vec{A} \) be \( x \) and the magnitude of \( \vec{B} \) be \( 2x \) (since the ratio is \( 1:2 \)).

**Step 2: Use the formula for the magnitude of the difference of two vectors.**
The magnitude of the difference of two vectors can be calculated using the formula:
\[ |\vec{B} - \vec{A}| = \sqrt{|\vec{B}|^2 + |\vec{A}|^2 - 2 |\vec{A}| |\vec{B}| \cos(\theta)} \]
where \( \theta \) is the angle between the two vectors.

**Step 3: Substitute the known values into the formula.**
Here, \( |\vec{A}| = x \), \( |\vec{B}| = 2x \), and \( \theta = 120^\circ \). The cosine of \( 120^\circ \) is \( -\frac{1}{2} \). Thus, we can write:
\[ |\vec{B} - \vec{A}| = 10 = \sqrt{(2x)^2 + x^2 - 2 \cdot x \cdot 2x \cdot \left(-\frac{1}{2}\right)} \]

**Step 4: Simplify the equation.**
Calculating the terms inside the square root:
\[ (2x)^2 = 4x^2 \]
\[ x^2 = x^2 \]
\[ -2 \cdot x \cdot 2x \cdot \left(-\frac{1}{2}\right) = 2x^2 \]
Now substituting these back into the equation gives:
\[ 10 = \sqrt{4x^2 + x^2 + 2x^2} = \sqrt{7x^2} \]

**Step 5: Square both sides to eliminate the square root.**
\[ 10^2 = 7x^2 \implies 100 = 7x^2 \]

**Step 6: Solve for \( x^2 \).**
\[ x^2 = \frac{100}{7} \]
\[ x = \sqrt{\frac{100}{7}} = \frac{10}{\sqrt{7}} \]

**Step 7: Find the magnitudes of \( \vec{A} \) and \( \vec{B} \).**
Now that we have \( x \):
- The magnitude of \( \vec{A} \) is:
\[ |\vec{A}| = x = \frac{10}{\sqrt{7}} \]
- The magnitude of \( \vec{B} \) is:
\[ |\vec{B}| = 2x = 2 \cdot \frac{10}{\sqrt{7}} = \frac{20}{\sqrt{7}} \]

### Final Answer:
The magnitudes of \( \vec{A} \) and \( \vec{B} \) are:
\[ |\vec{A}| = \frac{10}{\sqrt{7}} \quad \text{and} \quad |\vec{B}| = \frac{20}{\sqrt{7}} \]