Question

Let \(\overset{̄}{a} = \hat{i}+\hat{j}\), \(\overset{̄}{c} = \hat{i}-\hat{j}\) and a vector \(\overset{̄}{b}\) be such that \(\overset{̄}{a}\times \overset{̄}{b} = \overset{̄}{c}\) and \(\overset{̄}{a}\cdot \overset{̄}{b} = 3\) then \(|\overset{̄}{b}| =\)

• A. \(\frac{11}{\sqrt{2}}\)
• B. \(\frac{11}{\sqrt{3}}\)
• C. \(\sqrt{\frac{11}{2}}\)
• D. \(\sqrt{\frac{11}{3}}\)

Hint:

Take the cross product of a with both sides to get an expression for b.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Use the triple product identity \(\vec a\times(\vec a\times\vec b) = (\vec a\cdot\vec b)\vec a - (\vec a\cdot\vec a)\vec b\).

Step 2: Compute the left side with the given data:
\(\vec a\times\vec c = (\hat i + \hat j)\times(\hat i - \hat j) = -\hat k - \hat k = -2\hat k\). Here \(\vec a\cdot\vec a = 2\) and \(\vec a\cdot\vec b = 3\).
\[ 3(\hat i + \hat j) - 2\vec b = -2\hat k \]

Step 3: Solve for b:
\[ \vec b = \frac{3(\hat i + \hat j) + 2\hat k}{2} = \tfrac32\hat i + \tfrac32\hat j + \hat k \]

Step 4: Magnitude:
\[ |\vec b|^2 = \frac94 + \frac94 + 1 = \frac{11}{2} \Rightarrow |\vec b| = \sqrt{\frac{11}{2}} \]
Check: \(\vec a\times\vec b = \hat i - \hat j = \vec c\) and \(\vec a\cdot\vec b = 3\). Option (C).

Final Answer:
b = (3/2, 3/2, 1), so |b| = sqrt(11/2). \[ \boxed{\text{(C) }\sqrt{\dfrac{11}{2}}} \]