egin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ w_1 & w_2 & w_3 \ 3 & -2 & 1 \end{vmatrix} = \langle -1, 4, 2

["Understanding the Cross Product: Computation and Interpretation of (\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 3 & -2 & 1 \ w_1 & w_2 & w_3 \end{vmatrix} = \langle -1, 4, 2 \rangle)", "The cross product is a fundamental operation in vector algebra, widely used in physics, engineering, and computer graphics. One common application involves computing the cross product of a known vector with an unknown vector, resulting in a new vector expressed through its components in the standard basis. This article explores the mathematical computation and geometric significance of the identity:", "[\n\begin{vmatrix} \n\mathbf{i} & \mathbf{j} & \mathbf{k} \ \n3 & -2 & 1 \ \nw_1 & w_2 & w_3 \n\end{vmatrix} = \langle -1, 4, 2 \rangle\n]", "Our focus is on determining (w_1), (w_2), and (w_3) that satisfy this equation, while explaining the underlying linear algebra and vector properties.", "---", "### What Is a Cross Product?", "The cross product of two vectors (\mathbf{a} = \langle a_1, a_2, a_3 \rangle) and (\mathbf{b} = \langle b_1, b_2, b_3 \rangle) in (\mathbb{R}^3) is a vector defined as:", "[\n\mathbf{a} \ imes \mathbf{b} = \n\begin{vmatrix} \n\mathbf{i} & \mathbf{j} & \mathbf{k} \ \na_1 & a_2 & a_3 \ \nb_1 & b_2 & b_3 \n\end{vmatrix}\n= \langle a_2 b_3 - a_3 b_2, ; a_3 b_1 - a_1 b_3, ; a_1 b_2 - a_2 b_1 \rangle\n]", "This determinant expands how vectors interact spatially, producing a vector orthogonal to both input vectors.", "---", "### Applying the Determinant", "We compute the cross product using the given determinant:", "[\n\begin{vmatrix} \n\mathbf{i} & \mathbf{j} & \mathbf{k} \ \n3 & -2 & 1 \ \nw_1 & w_2 & w_3 \n\end{vmatrix}\n]", "Using the cofactor expansion along the first row:", "- (x)-component ((\mathbf{i})):\n (-2 \cdot w_3 - 1 \cdot w_2 = -2w_3 - w_2)", "- (y)-component ((\mathbf{j})):\n (1 \cdot w_1 - 3 \cdot w_3 = w_1 - 3w_3)\nNote: In standard conventions, this component is negated in vector form because the determinant expansion naturally assigns the sign via the orientation of columns — here, the negative sign is inherent, so it appears as is.", "- (z)-component ((\mathbf{k})):\n (3 \cdot w_2 - (-2) \cdot w_1 = 3w_2 + 2w_1)", "Putting it all together:", "[\n\mathbf{a} \ imes \mathbf{b} = \langle -2w_3 - w_2, ; w_1 - 3w_3, ; 2w_1 + 3w_2 \rangle\n]", "---", "### Equating Components to the Given Result", "We are told:", "[\n\langle -2w_3 - w_2, ; w_1 - 3w_3, ; 2w_1 + 3w_2 \rangle = \langle -1, 4, 2 \rangle\n]", "This gives us a system of three linear equations:", "1. (-2w_3 - w_2 = -1)       (1)\n2. ( w_1 - 3w_3 = 4 )        (2)\n3. ( 2w_1 + 3w_2 = 2 )       (3)", "We solve this system step by step.", "---", "### Step 1: Solve for (w_1) in terms of (w_3)", "From equation (2):\n[\nw_1 = 4 + 3w_3\n]", "---", "### Step 2: Express (w_2) from equation (1)", "Equation (1):\n[\n-2w_3 - w_2 = -1 \Rightarrow w_2 = -2w_3 + 1\n]", "---", "### Step 3: Substitute into equation (3)", "Plug (w_1 = 4 + 3w_3) and (w_2 = -2w_3 + 1) into equation (3):", "[\n2(4 + 3w_3) + 3(-2w_3 + 1) = 2\n]", "Simplify:", "[\n8 + 6w_3 - 6w_3 + 3 = 2\n\Rightarrow 8 + 3 = 2\n\Rightarrow 11 = 2\n]", "This is a contradiction.", "Wait — this suggests an inconsistency. But recall: cross product results must satisfy orthogonality. Let’s verify a fundamental property:", "The cross product (\mathbf{v} \ imes \mathbf{u}) is always orthogonal to both (\mathbf{v}) and (\mathbf{u}). Therefore, the given vector (\langle -1, 4, 2 \rangle) must be orthogonal to the result of the cross product operations — more critically, it must also satisfy:", "[\n\mathbf{c} \cdot (\mathbf{a} \ imes \mathbf{b}) = 0\n]", "where (\mathbf{c} = \langle 3, -2, 1 \rangle), and (\mathbf{v} = \langle w_1, w_2, w_3 \rangle).", "Compute:", "[\n\langle 3, -2, 1 \rangle \cdot \langle -1, 4, 2 \rangle = 3(-1) + (-2)(4) + 1(2) = -3 -8 + 2 = -9 <br/>\neq 0\n]", "This is impossible — the given vector is not orthogonal to the vector defining the unknowns, so it cannot be the cross product of (\langle 3, -2, 1 \rangle) with any vector (\langle w_1, w_2, w_3 \rangle).", "---", "### Re-evaluating the Problem", "Since the dot product does not vanish, no such vector (\mathbf{w} = \langle w_1, w_2, w_3 \rangle) exists that satisfies the equation:", "[\n\begin{vmatrix} \n\mathbf{i} & \mathbf{j} & \mathbf{k} \ \n3 & -2 & 1 \ \nw_1 & w_2 & w_3 \n\end{vmatrix} = \langle -1, 4, 2 \rangle\n]", "Instead, the correct condition for such an equation to hold is that the right-hand side must lie in the plane spanned by (\mathbf{i}, \mathbf{j}, \mathbf{k}) and be decomposable via a cross product — but more importantly, orthogonality with the first vector is mandatory.", "We conclude: There is an error in the provided identity as written — the vector (\langle -1, 4, 2 \rangle) is not orthogonal to (\langle 3, -2, 1 \rangle), so no real ((w_1, w_2, w_3)) satisfies the equation.", "---", "### Interpretation and Practical Advice", "When encountering cross product problems:", "- Always check orthogonality: (\mathbf{a} \ imes \mathbf{b} \cdot \mathbf{a} = 0).\n- Verify that computed results are consistent with vector space axioms.\n- Use determinants to compute cross products, but never assume arbitrary outcomes — the system must be consistent.", "---", "### Summary", "The computation technique — expanding the determinant using cofactor expansion — is correct and instructive for handling 3D vector operations. However, applying it to an impossible vector relationship exposes a flaw in the premise.", "If you seek the vector (\mathbf{w}) such that (\langle 3, -2, 1 \rangle \ imes \mathbf{w} = \mathbf{v}), solutions exist only if (\mathbf{v} \cdot \langle 3, -2, 1 \rangle = 0). Otherwise, no solution exists.", "For valid cross products, always verify orthogonality first.", "---", "Key Takeaways:", "- Cross products yield vectors orthogonal to the input vectors.\n- Any proposed cross product vector must satisfy orthogonality with the source vector.\n- Algebraic computation applies only when the system is consistent.\n- Double-check all vector identities and dot product conditions.", "---", "Related Topics:\nCross product properties, Vector space geometry, Determinant expansion, Linear systems in (\mathbb{R}^3), Orthogonality in (\mathbb{R}^3)", "For further reading, consult vector algebra textbooks or online resources on cross products and 3×3 determinants.", "---", "Keywords:\ncross product calculation, vector determinant expansion, (\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 3 & -2 & 1 \ w_1 & w_2 & w_3 \end{vmatrix} = \langle -1, 4, 2 \rangle), vector orthogonality, linear algebra, 3D vectors, cross product orthogonality condition"]