Let \(\mathbf{v} = egin{pmatrix} x \ y \ z \end{pmatrix}\), \(\mathbf{a} = egin{pmatrix} 1 \ 2 \ 3 \end{pmatrix}\), and \(\mathbf{v} imes \mathbf{a} = egin{pmatrix} 0 \ 0 \ 5 \end{pmatrix}\).
["Understanding the Cross Product: Solving (\mathbf{v} \ imes \mathbf{a} = \begin{pmatrix} 0 \ 0 \ 5 \end{pmatrix})", "The cross product of two vectors is a fundamental operation in linear algebra with wide applications in physics, engineering, and computer graphics. In this article, we explore the cross product (\mathbf{v} \ imes \mathbf{a}) where (\mathbf{v} = \begin{pmatrix} x \ y \ z \end{pmatrix}), (\mathbf{a} = \begin{pmatrix} 1 \ 2 \ 3 \end{pmatrix}), and the result equals (\begin{pmatrix} 0 \ 0 \ 5 \end{pmatrix}). We’ll break down the mathematical derivation, interpret its geometric meaning, and explain how to solve for the unknown components of (\mathbf{v}).", "---", "### What is the Cross Product?", "The cross product (\mathbf{v} \ imes \mathbf{a}) of two 3D vectors (\mathbf{v} = \begin{pmatrix} x \ y \ z \end{pmatrix}) and (\mathbf{a} = \begin{pmatrix} a \ b \ c \end{pmatrix}) is defined as:", "[\n\mathbf{v} \ imes \mathbf{a} = \n\begin{pmatrix}\ny \cdot 3 - z \cdot 2 \\nz \cdot 1 - x \cdot 3 \\nx \cdot 2 - y \cdot 1\n\end{pmatrix}\n= \n\begin{pmatrix}\n(3y - 2z) \\n(3z - 3x) \\n(2x - y)\n\end{pmatrix}\n]", "This vector is orthogonal (perpendicular) to both (\mathbf{v}) and (\mathbf{a}), and its magnitude equals the area of the parallelogram formed by (\mathbf{v}) and (\mathbf{a}).", "---", "### Given Information", "We are given:\n[\n\mathbf{v} = \begin{pmatrix} x \ y \ z \end{pmatrix},\quad\n\mathbf{a} = \begin{pmatrix} 1 \ 2 \ 3 \end{pmatrix},\quad\n\mathbf{v} \ imes \mathbf{a} = \begin{pmatrix} 0 \ 0 \ 5 \end{pmatrix}\n]", "Using the cross product formula derived above, equating components yields the system:", "1. (3y - 2z = 0)\n2. (3z - 3x = 0)\n3. (2x - y = 5)", "---", "### Solving the System of Equations", "Step 1: Simplify equations.", "From equation (2):\n[\n3z = 3x \Rightarrow z = x\n]", "From equation (1):\n[\n3y = 2z \Rightarrow y = \frac{2}{3}z\n]", "Substitute (z = x) and (y = \frac{2}{3}x) into equation (3):", "[\n2x - \left(\frac{2}{3}x\right) = 5 \Rightarrow \left(\frac{6x - 2x}{3}\right) = 5 \Rightarrow \frac{4x}{3} = 5\n]", "Solve for (x):", "[\n4x = 15 \Rightarrow x = \frac{15}{4}\n]", "Now compute (z) and (y):", "[\nz = x = \frac{15}{4},\quad y = \frac{2}{3} \cdot \frac{15}{4} = \frac{30}{12} = \frac{5}{2}\n]", "---", "### Final Solution", "The vector (\mathbf{v}) that satisfies the equation is:", "[\n\mathbf{v} = \begin{pmatrix}\n\dfrac{15}{4} \ \dfrac{5}{2} \ \dfrac{15}{4}\n\end{pmatrix}\n]", "---", "### Geometric Interpretation", "The result (\mathbf{v} \ imes \mathbf{a} = \begin{pmatrix} 0 \ 0 \ 5 \end{pmatrix}) points straight along the (z)-axis, indicating (\mathbf{v}) and (\mathbf{a}) lie in a plane perpendicular to this axis. The cross product’s magnitude ( | \mathbf{v} | \cdot | \mathbf{a} | \cdot \sin\ heta ) equals 5, where (\ heta) is the angle between (\mathbf{v}) and (\mathbf{a}).", "---", "### Practical Applications", "Understanding such vector equations is essential in:", "- Physics: Calculating torque, angular momentum, and magnetic forces.\n- Computer Graphics: Determining surface normals and rendering.\n- Robotics: Computing forces and motion constraints.", "---", "### Summary", "Solving (\mathbf{v} \ imes \mathbf{a} = \begin{pmatrix} 0 \ 0 \ 5 \end{pmatrix}) involves setting up a system from the cross product components and solving algebraically. In this case, (\mathbf{v}) is uniquely determined as (\begin{pmatrix} \dfrac{15}{4} \ \dfrac{5}{2} \ \dfrac{15}{4} \end{pmatrix}). Mastering cross products enables deeper insights into 3D spatial relationships and vector behavior.", "---", "Keywords: cross product, vector math, (\mathbf{v} \ imes \mathbf{a}), linear algebra, physics applications, vector components, (\begin{pmatrix} 0 \ 0 \ 5 \end{pmatrix})", "---", "This explanation provides a clear, step-by-step foundation for solving vector cross product problems and appreciating their physical meaning. Whether you're studying for math exams, coding in graphics, or engineering, mastering this technique is vital."]
