Correct Answer - Option 4 :
\(\frac{2\sqrt 2}{3\sqrt 7}\)
CONCEPT:
- \(\vec a \cdot \vec a = |\vec a|^2\)
- \(\vec a \cdot \vec b = \vec b \cdot \vec a\)
CALCULATION:
Given: \((\vec a + \vec b) \cdot (\vec a - \vec b) = 8 \ and \ |\vec a| = 8|\vec b|\)
⇒ \((\vec a + \vec b) \cdot (\vec a - \vec b) = |\vec a|^2 - \vec a \cdot \vec b + \vec b \cdot \vec a - |\vec b|^2 = 8\)
As we know that, \(\vec a \cdot \vec b = \vec b \cdot \vec a\)
⇒ \( |\vec a|^2 - |\vec b|^2 = 8\)
∵ It is given that \(|\vec a| = 8|\vec b|\)
⇒ \(63|\vec b|^2 = 8\)
⇒ \(|\vec b| = \frac{2\sqrt 2}{3\sqrt 7}\)
Hence, option 4 is correct.
