\mathbf{v} \times \mathbf{a} = \begin{pmatrix} y(4) - z(-1) \\ z(2) - x(4) \\ x(-1) - y(2) \end{pmatrix} = \begin{pmatrix} 4y + z \\ 2z - 4x \\ -x - 2y \end{pmatrix} = \begin{pmatrix} 5 \\ -6 \\ -1 \end{pmatrix}

["Understanding the Vector Equation: A Step-by-Step Breakdown of ( \mathbf{v} \ imes \mathbf{a} = \begin{pmatrix} 5 \ -6 \ -1 \end{pmatrix} )", "Vector cross products are powerful tools in mathematics, physics, and engineering—used commonly in 3D dynamics, electromagnetism, and computer graphics. Today, we explore a specific vector equation involving a cross product and solve for unknown vector components.", "---", "### What is This Vector Equation?", "We are given:", "[\n\mathbf{v} \ imes \mathbf{a} = \begin{pmatrix} 5 \ -6 \ -1 \end{pmatrix}\n]", "And after expansion, the left-hand side simplifies to:", "[\n\mathbf{v} \ imes \mathbf{a} = \begin{pmatrix} y(4) - z(-1) \ z(2) - x(4) \ x(-1) - y(2) \end{pmatrix} = \begin{pmatrix} 4y + z \ 2z - 4x \ -x - 2y \end{pmatrix}\n]", "So the full vector equation becomes:", "[\n\begin{pmatrix} 4y + z \ 2z - 4x \ -x - 2y \end{pmatrix} = \begin{pmatrix} 5 \ -6 \ -1 \end{pmatrix}\n]", "---", "### Step 1: Translate Component Equations", "From vector equality, we derive a system of linear equations:", "1. ( 4y + z = 5 )  (Equation 1)\n2. ( 2z - 4x = -6 )  (Equation 2)\n3. ( -x - 2y = -1 )  (Equation 3)", "---", "### Step 2: Solve the System of Equations", "We will solve for ( x ), ( y ), and ( z ) to interpret ( \mathbf{v} ).", "Equation 3:\n[\n-x - 2y = -1 \implies x + 2y = 1 \quad \ ext{(1)}\n]", "Equation 2:\n[\n2z - 4x = -6 \implies z - 2x = -3 \quad \ ext{(2a)}\n]\nFrom (1): ( x = 1 - 2y )\nSubstitute into (2a):", "[\nz - 2(1 - 2y) = -3 \implies z - 2 + 4y = -3 \implies z = -1 - 4y \quad \ ext{(2b)}\n]", "Equation 1:\n[\n4y + z = 5\n]\nSubstitute ( z ) from (2b):", "[\n4y + (-1 - 4y) = 5 \implies -1 = 5\n]", "Wait—this gives (-1 = 5), a contradiction.", "---", "### Step 3: Re-examining the Equations", "A contradiction implies inconsistency. But since the cross product is defined uniquely given vectors ( \mathbf{v} ) and ( \mathbf{a} ), the system must be consistent.", "Let’s double-check derivation.", "The cross product:", "If ( \mathbf{v} = \begin{pmatrix} v_1 \ v_2 \ v_3 \end{pmatrix} ), and from prior expansion,", "[\n\mathbf{v} \ imes \mathbf{a} \propto \begin{pmatrix} v_2 a_3 - v_3 a_2 \ v_3 a_1 - v_1 a_3 \ v_1 a_2 - v_2 a_1 \end{pmatrix}\n]", "Matching components, we see:", "- First component: ( v_2 a_3 - v_3 a_2 = 4y + z )\n- Second: ( v_3 a_1 - v_1 a_3 = 2z - 4x )\n- Third: ( v_1 a_2 - v_2 a_1 = -x - 2y )", "This suggests the components of ( \mathbf{a} ) are tied to ( x, y, z )—but perhaps ( \mathbf{a} ) is a point? Or ( \mathbf{v} ) is variable?", "But in the equation given, all components depend only on variables ( x, y, z ), not on components of ( \mathbf{a} ). So reconsider: likely ( \mathbf{a} ) is not a vector, but a fixed vector or constant?", "Alternatively, perhaps the cross product is expressed in terms of the components of some fixed vector ( \mathbf{a} )—but our earlier assumption leads to contradiction.", "Re-analyze: the expression", "[\n\mathbf{v} \ imes \mathbf{a} = \begin{pmatrix} 4y + z \ 2z - 4x \ -x - 2y \end{pmatrix}\n]", "suggests the cross product depends linearly on variables ( x, y, z )—not on components of ( \mathbf{a} ). Therefore, this is not a standard cross product ( \mathbf{v} \ imes \mathbf{a} ) unless ( \mathbf{a} ) is expressed via ( x, y, z ), e.g., ( \mathbf{a} = \begin{pmatrix} 4 \ 0 \ -1 \end{pmatrix} ) or something.", "But let’s suppose the equation is:", "[\n\mathbf{v} \ imes \mathbf{a} = \mathbf{w}, \quad \ ext{where } \mathbf{w} = \begin{pmatrix} 5 \ -6 \ -1 \end{pmatrix}\n]", "and the expansion in terms of variables ( x, y, z ) suggests ( \mathbf{a} ) has components in terms of ( x, y, z ), or vice versa.", "However, the format clearly shows the RHS depends only on ( x, y, z ) linearly—suggesting this is not ( \mathbf{v} \ imes \mathbf{a} ) with fixed ( \mathbf{a} ), but rather a linear system arising from cross product components.", "Assume instead: this is a system derived from a cross product, and we must solve for ( \mathbf{v} = (v_1, v_2, v_3) ), while ( \mathbf{a} ) is unknown—or conversely, ( \mathbf{a} ) is a point.", "But given the data, the cleanest interpretation is that the system:", "[\n\begin{cases}\n4y + z = 5 \\n-4x + 2z = -6 \\n- x - 2y = -1\n\end{cases}\n]", "defines a linear system in variables ( x, y, z )—but that contradicts cross product structure unless ( \mathbf{a} ) is defined via these.", "Wait—perhaps the cross product is such that the coefficients are given asymptotically? That seems unlikely.", "Alternatively, consider this: this system may represent the components of the cross product vector in a coordinate system tied to ( x, y, z ), but that is highly non-standard.", "Let’s shift perspective: perhaps the equation is identically satisfied only when the vector equation holds, and we are to solve for ( x, y, z ).", "Therefore, accept:", "We solve:", "[\n\begin{aligned}\n4y + z &= 5 \quad \ ext{(1)}\\n-4x + 2z &= -6 \quad \ ext{(2)}\\n- x - 2y &= -1 \quad \ ext{(3)}\n\end{aligned}\n]", "Solve step-by-step.", "From (3):\n[\nx = -2y + 1\n]", "Plug into (2):\n[\n-4(-2y + 1) + 2z = -6 \implies 8y - 4 + 2z = -6 \implies 2z = -2 - 8y \implies z = -1 - 4y \quad \ ext{(2')}\n]", "Now plug into (1):\n[\n4y + (-1 - 4y) = 5 \implies -1 = 5\n]", "Contradiction again.", "---", "### Step 4: Identify the Issue — Is the RHS Consistent?", "The contradiction ( -1 = 5 ) implies no solution exists for ( x, y, z ) satisfying the system—unless the RHS is inconsistent.", "But the problem presents the equation as valid.", "Therefore, likely: the given cross product identity is not for variables ( x, y, z ), but rather ( \mathbf{v} \ imes \mathbf{a} = \mathbf{w} ), and ( \mathbf{w} ) is expressed in terms of variables indexed with ( x, y, z )—but what are ( x, y, z ) indices?", "Reconsider the notation: in some contexts, ( y(4) ) means "the component of ( y )"]