Question

If \(\overset{̄}{a}\cdot \overset{̄}{b} = β\) and \(\overset{̄}{a}\times \overset{̄}{b} = \overset{̄}{c}\) then \(\overset{̄}{a} =\)

• A. \(\frac{\overset{̄}{b}\times \overset{̄}{c}-β\overset{̄}{b}}{|\overset{̄}{b}|^2}\)
• B. \(\frac{\overset{̄}{b}\times \overset{̄}{c}-β\overset{̄}{c}}{|\overset{̄}{b}|^2}\)
• C. \(\frac{\overset{̄}{b}\times \overset{̄}{c}+β\overset{̄}{b}}{|\overset{̄}{b}|^2}\)
• D. \(\frac{\overset{̄}{b}\times \overset{̄}{c}+β\overset{̄}{c}}{|\overset{̄}{b}|^2}\)

Hint:

Compute b x c = b x (a x b) using the triple product expansion.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We know \(\bar a\cdot\bar b = \beta\) and \(\bar a\times\bar b = \bar c\). Crossing \(\bar b\) with \(\bar c\) brings out \(\bar a\) through the triple product rule.

Step 2: Key Formula or Approach:
\[ \bar b\times(\bar a\times\bar b) = \bar a(\bar b\cdot\bar b) - \bar b(\bar b\cdot\bar a) \]

Step 3: Detailed Explanation:
Substitute \(\bar c = \bar a\times\bar b\):
\[ \bar b\times\bar c = \bar b\times(\bar a\times\bar b) = \bar a|\bar b|^2 - \bar b\,(\bar a\cdot\bar b) = \bar a|\bar b|^2 - \beta\bar b \]
Solve for \(\bar a\):
\[ \bar a|\bar b|^2 = \bar b\times\bar c + \beta\bar b \]
\[ \bar a = \frac{\bar b\times\bar c + \beta\bar b}{|\bar b|^2} \]
Options (A) and (B) have a minus sign before \(\beta\bar b\), which would be the case if the term was moved incorrectly. Option (D) uses \(\beta\bar c\), but \(\bar c\) is perpendicular to \(\bar b\) and cannot be multiplied by a dot product to rebuild the part of \(\bar a\) along \(\bar b\).

Final Answer:
\(\bar a = \dfrac{\bar b\times\bar c + \beta\bar b}{|\bar b|^2}\), option (C). \[ \boxed{\frac{\bar b\times\bar c+\beta\bar b}{|\bar b|^2} \text{ (C)}} \]