Solution: The vector equation $\mathbf{v} imes \mathbf{a} = \mathbf{b}$ has a solution only if $\mathbf{a} \cdot \mathbf{b} = 0$, since the cross product is perpendicular to $\mathbf{a}$.
["Understanding Why $\mathbf{v} \ imes \mathbf{a} = \mathbf{b}$ Requires $\mathbf{a} \cdot \mathbf{b} = 0$", "In vector algebra, the equation $\mathbf{v} \ imes \mathbf{a} = \mathbf{b}$ arises frequently in physics and engineering, especially in problems involving torque, angular momentum, and rotational motion. A fundamental condition for this cross product equation to have a solution is that $\mathbf{a} \cdot \mathbf{b} = 0$. But why is this the case? Let’s explore the mathematical reasoning and significance behind this condition.", "### The Cross Product Is Perpendicular to Both Vectors", "First, recall a key property of the cross product: the vector $\mathbf{v} \ imes \mathbf{a}$ is always perpendicular to both $\mathbf{v}$ and $\mathbf{a}$. Mathematically, this means:", "$$\n\mathbf{v} \ imes \mathbf{a} \cdot \mathbf{a} = 0\n$$", "Using the distributive property of the dot product, we rewrite this as:", "$$\n\mathbf{a} \cdot (\mathbf{v} \ imes \mathbf{a}) = 0\n$$", "But the left-hand side is precisely the scalar triple product $\mathbf{a} \cdot (\mathbf{v} \ imes \mathbf{a})$, which equals $\mathbf{b} \cdot \mathbf{a}$ when $\mathbf{b} = \mathbf{v} \ imes \mathbf{a}$. Therefore:", "$$\n\mathbf{a} \cdot \mathbf{b} = 0\n$$", "This geometric insight reveals that $\mathbf{b}$ must lie in a plane perpendicular to $\mathbf{a}$. If $\mathbf{a}$ and $\mathbf{b}$ are not perpendicular, no vector $\mathbf{v}$ can satisfy $\mathbf{v} \ imes \mathbf{a} = \mathbf{b}$, since the cross product can never align with or span a direction parallel to $\mathbf{a}$.", "### Geometric Interpretation and Feasibility Condition", "Geometrically, the cross product $\mathbf{v} \ imes \mathbf{a}$ generates a vector orthogonal to both $\mathbf{v}$ and $\mathbf{a}$. For this vector to equal $\mathbf{b}$, $\mathbf{b}$ must lie in the plane formed by $\mathbf{v}$ and $\mathbf{a}$. But if $\mathbf{a} \cdot \mathbf{b} <br/>\ne 0$, $\mathbf{b}$ points in a direction that is strictly perpendicular to $\mathbf{a}$, which contradicts the orthogonality requirement enforced by the cross product.", "Thus, $\mathbf{a} \cdot \mathbf{b} = 0$ is both a mathematical necessity and a geometric constraint ensuring consistency in the vector relationship.", "### Solving for $\mathbf{v}$ When the Condition Holds", "When $\mathbf{a} \cdot \mathbf{b} = 0$, the equation $\mathbf{v} \ imes \mathbf{a} = \mathbf{b}$ is feasible. Solutions for $\mathbf{v}$ can then be found using vector algebra identities. One way to express $\mathbf{v}$ is:", "$$\n\mathbf{v} = \frac{\mathbf{a} \ imes \mathbf{b}}{|\mathbf{a}|^2} + k\mathbf{a}, \quad \ ext{where } k \ ext{ is any scalar}\n$$", "This general solution reflects that any component of $\mathbf{v}$ parallel to $\mathbf{a}$ (the $k\mathbf{a}$ term) does not affect the cross product, while the orthogonal component $\frac{\mathbf{a} \ imes \mathbf{b}}{|\mathbf{a}|^2}$ captures the non-parallel part required to produce $\mathbf{b}$ when crossed with $\mathbf{a}$.", "---", "Summary", "- The cross product $\mathbf{v} \ imes \mathbf{a} = \mathbf{b}$ has a solution only if $\mathbf{a} \cdot \mathbf{b} = 0$.\n- This condition arises because the cross product $\mathbf{v} \ imes \mathbf{a}$ is always perpendicular to $\mathbf{a}$.\n- Geometrically, $\mathbf{b}$ must lie in the plane orthogonal to $\mathbf{a}$ for the equation to hold.\n- When satisfied, the solution for $\mathbf{v}$ includes a particular part perpendicular to $\mathbf{a}$ and an arbitrary parallel component, forming a general solution.", "Understanding this condition ensures correct modeling and computation in physics and engineering applications involving vector cross products."]
