The vector cross product \(\mathbf{v} \times \mathbf{b} = \langle v_1, v_2, v_3 \rangle \times \langle 0, 1, 0 \rangle\) is computed as:

["The Vector Cross Product (\mathbf{v} \ imes \mathbf{b} = \langle v_1, v_2, v_3 \rangle \ imes \langle 0, 1, 0 \rangle) Explained Step-by-Step", "Cross products are a fundamental operation in vector mathematics, especially in physics and engineering, helping define quantities like torque, angular momentum, and rotational motion. When computing the cross product (\mathbf{v} \ imes \mathbf{b}), where (\mathbf{v} = \langle v_1, v_2, v_3 \rangle) and (\mathbf{b} = \langle 0, 1, 0 \rangle), the result is a vector that combines well-defined algebraic rules with geometric intuition. This article explains how and why this cross product is computed, providing clarity on the mechanics and significance of each component.", "---", "### Understanding the Cross Product Concept", "The cross product of two vectors in three-dimensional space produces a third vector perpendicular to both original vectors. If you have two vectors (\mathbf{v} = \langle v_1, v_2, v_3 \rangle) and (\mathbf{b} = \langle 0, 1, 0 \rangle), the cross product (\mathbf{v} \ imes \mathbf{b}) follows a specific determinant-based formula.", "---", "### The Formula for Cross Product", "The cross product can be computed using the following determinant:", "[\n\mathbf{v} \ imes \mathbf{b} = \n\begin{vmatrix}\n\mathbf{i} & \mathbf{j} & \mathbf{k} \\nv_1 & v_2 & v_3 \\n0 & 1 & 0 \\n\end{vmatrix}\n= \mathbf{i}(v_2 \cdot 0 - v_3 \cdot 1) - \mathbf{j}(v_1 \cdot 0 - v_3 \cdot 0) + \mathbf{k}(v_1 \cdot 1 - v_2 \cdot 0)\n]", "Simplifying component-wise:", "-\nThe (\mathbf{i})-component: (v_2 \cdot 0 - v_3 \cdot 1 = -v_3)", "-\nThe (\mathbf{j})-component:\n(u-process; note (-\mathbf{j}(0 - 0) = 0))", "-\nThe (\mathbf{k})-component: (v_1 \cdot 1 - v_2 \cdot 0 = v_1)", "So, the result becomes:", "[\n\mathbf{v} \ imes \mathbf{b} = \langle -v_3, 0, v_1 \rangle\n]", "---", "### Step-by-Step Computation", "1. Set up the determinant with unit vectors:\n[\n\begin{vmatrix}\n\mathbf{i} & \mathbf{j} & \mathbf{k} \\nv_1 & v_2 & v_3 \\n0 & 1 & 0 \\n\end{vmatrix}\n]", "2. Expand along the third row:\nOnly the entries in the second row contribute because the others are zero:", "-\n[\n-i(0 - v_3 \cdot 1) = -i(-v_3) = \mathbf{i}(-v_3)\n]", "-\n[\n-j(0 - 0) = 0\n]", "-\n[\nk(v_1 \cdot 1 - v_2 \cdot 0) = k(v_1) = \mathbf{k}(v_1)\n]", "3. Combine components:\n[\n\mathbf{v} \ imes \mathbf{b} = \langle -v_3, 0, v_1 \rangle\n]", "---", "### Result Interpretation", "The cross product you compute:", "[\n\mathbf{v} \ imes \langle 0, 1, 0 \rangle = \langle -v_3, 0, v_1 \rangle\n]", "- The result is always perpendicular to both (\mathbf{v}) and (\mathbf{b}). Since (\mathbf{b} = \langle 0, 1, 0 \rangle) points in the (y)-direction, the output vector lies in the (xz)-plane.\n- The (y)-component vanishes, illustrating how the cross product enforces orthogonality: any common direction is nullified.\n- The components directly reflect the original vector’s (v_1) and (v_3) values, flipped and signed appropriately.", "---", "### Practical Applications", "This type of cross product regularly appears in physics:", "- Torque calculation: When a force (\mathbf{v}) is applied at a position (\mathbf{b}), the resulting torque vector (\mathbf{v} \ imes \mathbf{b}) determines rotational effect.\n- Angular velocity and linear momentum: Cross products help relate rotational and translational motion in mechanics.\n- Computer graphics and simulation: Defining directional outputs like normals and rotational forces relies on accurate cross product computation.", "---", "### Summary", "Computing (\mathbf{v} \ imes \langle 0, 1, 0 \rangle) via the cross product determinant reveals a precise, algebraic method to find a perpendicular vector encoding key geometric relationships. The result (\langle -v_3, 0, v_1 \rangle) serves as a powerful building block in vector calculus, physics, and engineering, illustrating how vectors interact in three-dimensional space.", "Understanding this operation deepens insight into spatial reasoning, rotational dynamics, and the mathematical foundation of forces and motion. Whether solving equilibrium problems or simulating physical systems, mastering the cross product is essential.", "---", "Keywords: vector cross product, (\mathbf{v} \ imes \mathbf{b}), (\langle v_1, v_2, v_3 \rangle \ imes \langle 0, 1, 0 \rangle), cross product formula, physics applications, torque, angular momentum, vector algebra.", "Meta Description:\nDiscover how the cross product (\mathbf{v} \ imes \langle 0, 1, 0 \rangle) is computed using determinants, what the resulting vector represents, and why it matters in physics and engineering applications. Learn step-by-step with clear algebra and interpretation."]