We are given that \( |\vec{a}| = 2|\vec{b}| \). To find \( |\vec{a} - \vec{b}| \), we use the following approach:
Step 1: Formula for the magnitude of the difference between two vectors:
\[ |\vec{a} - \vec{b}| = \sqrt{|\vec{a}|^2 + |\vec{b}|^2 - 2\vec{a} \cdot \vec{b}}. \]
Step 2: Substitute the relation \( |\vec{a}| = 2|\vec{b}| \):
Since \( |\vec{a}| = 2|\vec{b}| \), we have: \[ |\vec{a}|^2 = 4|\vec{b}|^2. \]
Step 3: Compute the dot product \( \vec{a} \cdot \vec{b} \):
The vectors \( \vec{a} \) and \( \vec{b} \) are in the same direction, so: \[ \vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}| = 2|\vec{b}|^2. \]
Step 4: Substitute these values into the magnitude formula:
\[ |\vec{a} - \vec{b}| = \sqrt{4|\vec{b}|^2 + |\vec{b}|^2 - 2 \cdot 2|\vec{b}|^2}. \]
Step 5: Simplify the expression:
\[ |\vec{a} - \vec{b}| = \sqrt{4|\vec{b}|^2 + |\vec{b}|^2 - 4|\vec{b}|^2} = \sqrt{|\vec{b}|^2}. \]
Step 6: Final computation:
\[ |\vec{a} - \vec{b}| = |\vec{b}| = 3. \]
Thus, the correct answer is:
\[ \boxed{3}. \]
List-I | List-II | ||
A | Megaliths | (I) | Decipherment of Brahmi and Kharoshti |
B | James Princep | (II) | Emerged in first millennium BCE |
C | Piyadassi | (III) | Means pleasant to behold |
D | Epigraphy | (IV) | Study of inscriptions |