Key Concepts and Formulas

• Geometric Interpretation of Cross Product: The cross product of two vectors, $\overrightarrow b \times \overrightarrow c$, results in a vector that is perpendicular to both $\overrightarrow b$ and $\overrightarrow c$.
• Perpendicularity Condition: If a vector $\overrightarrow a$ is the result of a cross product $\overrightarrow b \times \overrightarrow c$, then $\overrightarrow a$ must be perpendicular to $\overrightarrow c$. This means their dot product is zero: $\overrightarrow a \cdot \overrightarrow c = 0$.
• Dot Product Calculation: For vectors $\overrightarrow u = u_x \widehat i + u_y \widehat j + u_z \widehat k$ and $\overrightarrow v = v_x \widehat i + v_y \widehat j + v_z \widehat k$, their dot product is $\overrightarrow u \cdot \overrightarrow v = u_x v_x + u_y v_y + u_z v_z$.

Step-by-Step Solution

Step 1: Understand the Problem and Given Information We are given two vectors, $\overrightarrow a = \widehat i + \widehat j - \widehat k$ and $\overrightarrow c = 2\widehat i - 3\widehat j + 2\widehat k$. We need to find the number of vectors $\overrightarrow b$ that satisfy the equation $\overrightarrow b \times \overrightarrow c = \overrightarrow a$, with the additional constraint that the magnitude of $\overrightarrow b$, $|\overrightarrow b |$, must be an integer between 1 and 10, inclusive.

Step 2: Apply the Geometric Property of the Cross Product The fundamental property of the cross product states that if $\overrightarrow a = \overrightarrow b \times \overrightarrow c$, then the vector $\overrightarrow a$ must be perpendicular to both $\overrightarrow b$ and $\overrightarrow c$. A necessary condition for the existence of such a vector $\overrightarrow b$ is that $\overrightarrow a$ must be perpendicular to $\overrightarrow c$.

• Why this step? This is a crucial first check. If $\overrightarrow a$ is not perpendicular to $\overrightarrow c$, then no such vector $\overrightarrow b$ can exist, and we can immediately conclude that the number of solutions is zero, without needing to perform further calculations to find $\overrightarrow b$.

Step 3: Check the Perpendicularity Condition using the Dot Product For $\overrightarrow a$ and $\overrightarrow c$ to be perpendicular, their dot product $\overrightarrow a \cdot \overrightarrow c$ must be equal to zero. Let's calculate this dot product using the given vectors: $\overrightarrow a = 1\widehat i + 1\widehat j - 1\widehat k$ $\overrightarrow c = 2\widehat i - 3\widehat j + 2\widehat k$ $\overrightarrow a \cdot \overrightarrow c = (1)(2) + (1)(-3) + (-1)(2)$ $\overrightarrow a \cdot \overrightarrow c = 2 - 3 - 2$ $\overrightarrow a \cdot \overrightarrow c = -3$

Step 4: Analyze the Result of the Dot Product We found that $\overrightarrow a \cdot \overrightarrow c = -3$. According to the necessary condition established in Step 2, for a vector $\overrightarrow b$ to exist such that $\overrightarrow b \times \overrightarrow c = \overrightarrow a$, it must be true that $\overrightarrow a \cdot \overrightarrow c = 0$.

• Why this analysis? We are comparing the calculated dot product with the required condition. Since $-3 \neq 0$, the necessary condition for the existence of $\overrightarrow b$ is not satisfied. This means that the given vectors $\overrightarrow a$ and $\overrightarrow c$ are not perpendicular, and therefore, there is no vector $\overrightarrow b$ that can satisfy the equation $\overrightarrow b \times \overrightarrow c = \overrightarrow a$.

Step 5: Conclude the Number of Vectors $\overrightarrow b$ Because the fundamental requirement for the existence of $\overrightarrow b$ is violated, no such vector $\overrightarrow b$ exists, regardless of its magnitude. Therefore, the number of vectors $\overrightarrow b$ satisfying the given conditions is 0. The constraint on $|\overrightarrow b |$ becomes irrelevant in this case.

Common Mistakes & Tips

• Forgetting the Necessary Condition: A common mistake is to immediately try to find $\overrightarrow b$ without first checking if $\overrightarrow a \cdot \overrightarrow c = 0$. If this check is skipped, one might embark on a lengthy calculation that ultimately leads to a contradiction or no solution.
• Confusing Magnitude and Vector Existence: The problem asks for the number of vectors $\overrightarrow b$. If no vector $\overrightarrow b$ satisfies the primary vector equation, then the constraint on its magnitude is moot.
• Understanding Scalar Triple Product: The expression $\overrightarrow c \cdot (\overrightarrow b \times \overrightarrow c)$ is a scalar triple product. If you were to multiply both sides of $\overrightarrow b \times \overrightarrow c = \overrightarrow a$ by $\overrightarrow c$ using a dot product, you would get $\overrightarrow c \cdot (\overrightarrow b \times \overrightarrow c) = \overrightarrow c \cdot \overrightarrow a$. Since $\overrightarrow c \cdot (\overrightarrow b \times \overrightarrow c) = 0$ (as $\overrightarrow b \times \overrightarrow c$ is perpendicular to $\overrightarrow c$), this immediately leads to $\overrightarrow c \cdot \overrightarrow a = 0$, which is the same condition we derived from the geometric interpretation.

Summary

The problem requires finding the number of vectors $\overrightarrow b$ satisfying $\overrightarrow b \times \overrightarrow c = \overrightarrow a$ and a condition on $|\overrightarrow b|$. The key insight is that for such an equation to hold, the resultant vector $\overrightarrow a$ must be perpendicular to the vector $\overrightarrow c$. We tested this condition by calculating the dot product $\overrightarrow a \cdot \overrightarrow c$. Since $\overrightarrow a \cdot \overrightarrow c = -3 \neq 0$, the necessary perpendicularity condition is not met. Consequently, no vector $\overrightarrow b$ can exist that satisfies the given cross product equation. Therefore, the number of such vectors is zero.

The final answer is $\boxed{0}$ which corresponds to option (A).