Key Concepts and Formulas

• Vector Identity: For any two vectors $\overrightarrow u$ and $\overrightarrow v$, the following identity holds: $|\overrightarrow u \times \overrightarrow v|^2 + (\overrightarrow u \cdot \overrightarrow v)^2 = |\overrightarrow u|^2 |\overrightarrow v|^2$ This identity is derived from the geometric definitions of the dot product ($\overrightarrow u \cdot \overrightarrow v = |\overrightarrow u| |\overrightarrow v| \cos \theta$) and the magnitude of the cross product ($|\overrightarrow u \times \overrightarrow v| = |\overrightarrow u| |\overrightarrow v| \sin \theta$), where $\theta$ is the angle between the vectors.
• Magnitude of a Vector: For a vector $\overrightarrow v = a\widehat i + b\widehat j + c\widehat k$, its magnitude is given by $|\overrightarrow v| = \sqrt{a^2 + b^2 + c^2}$.

Step-by-Step Solution

Step 1: Calculate the magnitudes of the known vectors. We are given $\overrightarrow a = \widehat i + \widehat j + \widehat k$ and $\overrightarrow c = \widehat j - \widehat k$. We need their magnitudes to use in the vector identity.

• Magnitude of $\overrightarrow a$: $|\overrightarrow a| = \sqrt{(1)^2 + (1)^2 + (1)^2} = \sqrt{1 + 1 + 1} = \sqrt{3}$ Why this step? The vector identity requires $|\overrightarrow a|^2$, so calculating $|\overrightarrow a|$ is the first step.

• Magnitude of $\overrightarrow c$: $\overrightarrow c$ can be written as $0\widehat i + 1\widehat j - 1\widehat k$. $|\overrightarrow c| = \sqrt{(0)^2 + (1)^2 + (-1)^2} = \sqrt{0 + 1 + 1} = \sqrt{2}$ Why this step? We are given that $\overrightarrow a \times \overrightarrow b = \overrightarrow c$. Therefore, the magnitude of the cross product, $|\overrightarrow a \times \overrightarrow b|$, is equal to $|\overrightarrow c|$. This gives us the value needed for the term $|\overrightarrow a \times \overrightarrow b|^2$ in the identity.

Step 2: Apply the vector identity. We are given the identity $|\overrightarrow a \times \overrightarrow b|^2 + (\overrightarrow a \cdot \overrightarrow b)^2 = |\overrightarrow a|^2 |\overrightarrow b|^2$. We have the following information:

• $|\overrightarrow a \times \overrightarrow b| = |\overrightarrow c| = \sqrt{2}$, so $|\overrightarrow a \times \overrightarrow b|^2 = (\sqrt{2})^2 = 2$.
• $\overrightarrow a \cdot \overrightarrow b = 3$, so $(\overrightarrow a \cdot \overrightarrow b)^2 = (3)^2 = 9$.
• $|\overrightarrow a| = \sqrt{3}$, so $|\overrightarrow a|^2 = (\sqrt{3})^2 = 3$.
• $|\overrightarrow b|$ is what we need to find.

Substitute these values into the identity: $2 + 9 = 3 |\overrightarrow b|^2$ Why this step? This identity directly relates the magnitudes and dot/cross products of vectors, allowing us to form an equation to solve for the unknown magnitude $|\overrightarrow b|$.

Step 3: Solve for $|\overrightarrow b|$. Simplify the equation from Step 2: $11 = 3 |\overrightarrow b|^2$ Now, isolate $|\overrightarrow b|^2$: $|\overrightarrow b|^2 = \frac{11}{3}$ Finally, take the square root to find $|\overrightarrow b|$. Since magnitude must be non-negative, we take the positive root: $|\overrightarrow b| = \sqrt{\frac{11}{3}}$ Why this step? This is the final algebraic step to isolate and calculate the magnitude of vector $\overrightarrow b$.

Common Mistakes & Tips

• Forgetting the Identity: The identity $|\overrightarrow u \times \overrightarrow v|^2 + (\overrightarrow u \cdot \overrightarrow v)^2 = |\overrightarrow u|^2 |\overrightarrow v|^2$ is crucial for efficiently solving problems involving both dot and cross products.
• Confusing Magnitude and Magnitude Squared: Ensure you are using the correct terms in the identity. For example, $|\overrightarrow a \times \overrightarrow b|$ is $\sqrt{2}$, but the term in the identity is $|\overrightarrow a \times \overrightarrow b|^2$, which is $2$.
• Algebraic Errors: Double-check calculations when squaring numbers and when rearranging the equation to solve for $|\overrightarrow b|$.

Summary

This problem is efficiently solved by applying the fundamental vector identity relating the dot product, cross product, and magnitudes of two vectors. We first calculated the magnitudes of the given vectors $\overrightarrow a$ and $\overrightarrow c$. Using the given relationships $\overrightarrow a \times \overrightarrow b = \overrightarrow c$ and $\overrightarrow a \cdot \overrightarrow b = 3$, we substituted these values into the identity $|\overrightarrow a \times \overrightarrow b|^2 + (\overrightarrow a \cdot \overrightarrow b)^2 = |\overrightarrow a|^2 |\overrightarrow b|^2$. This allowed us to directly solve for $|\overrightarrow b|^2$ and subsequently for $|\overrightarrow b|$.

The final answer is $\boxed{\sqrt {{{11} \over 3}} }$, which corresponds to option (C).