Question

If \((\vec{a} - \vec{b}) \cdot (\vec{a} + \vec{b}) = 27\) and \(|\vec{a}| = 2|\vec{b}|\), then \(|\vec{b}|\) is:

• A. 3
• B. 2
• C. \( \frac{5}{6} \)
• D. 6

Answer 1

The Correct Option is A

We are given that \( |\vec{a}| = 2|\vec{b}| \). To find \( |\vec{a} - \vec{b}| \), we use the following approach: 
First, recall the formula for the magnitude of the difference between two vectors:
\[ |\vec{a} - \vec{b}| = \sqrt{|\vec{a}|^2 + |\vec{b}|^2 - 2\vec{a} \cdot \vec{b}}. \]
Since \( |\vec{a}| = 2|\vec{b}| \), we have:
\[ |\vec{a}|^2 = 4|\vec{b}|^2. \]
Now, let’s compute the dot product \( \vec{a} \cdot \vec{b} \). The vectors \( \vec{a} \) and \( \vec{b} \) are in the same direction, so \( \vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}| = 2|\vec{b}|^2 \).
Now substitute these values into the magnitude formula:
\[ |\vec{a} - \vec{b}| = \sqrt{4|\vec{b}|^2 + |\vec{b}|^2 - 2 \cdot 2|\vec{b}|^2}. \]
Simplifying:
\[ |\vec{a} - \vec{b}| = \sqrt{4|\vec{b}|^2 + |\vec{b}|^2 - 4|\vec{b}|^2} = \sqrt{|\vec{b}|^2} = |\vec{b}|. \]
Since \( |\vec{b}| = 3 \), we get:
\[ |\vec{a} - \vec{b}| = 3. \]
Thus, the correct answer is: 3.

Answer 2

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:

\[ \boxed{3}. \]