= \mathbf{i}(2 \cdot 4 - (-1) \cdot 1) - \mathbf{j}(1 \cdot 4 - (-1)(-2)) + \mathbf{k}(1 \cdot 1 - 2 \cdot (-2))

["Understanding the Vector Expression: Simplifying = $ \mathbf{i}(2 \cdot 4 - (-1) \cdot 1) - \mathbf{j}(1 \cdot 4 - (-1)(-2)) + \mathbf{k}(1 \cdot 1 - 2 \cdot (-2)) $", "Mathematics often expresses complex ideas in compact vector forms, and one such clean but powerful vector expression is:", "$$\n\mathbf{i}(2 \cdot 4 - (-1) \cdot 1) - \mathbf{j}(1 \cdot 4 - (-1)(-2)) + \mathbf{k}(1 \cdot 1 - 2 \cdot (-2))\n$$", "At first glance, this expression may look abstract, but it breaks down cleanly into real number components — a fundamental concept in vector algebra. In this article, we’ll decode each term, simplify the expression step-by-step, and explore how it connects to 3D vector geometry and operations.", "---", "### What Is This Vector All About?", "The expression states:", "$$\n\mathbf{v} = \mathbf{i} \underbrace{(2 \cdot 4 - (-1) \cdot 1)}{\ ext{i-component}} - \mathbf{j} \underbrace{(1 \cdot 4 - (-1)(-2))}}} + \mathbf{k} \underbrace{(1 \cdot 1 - 2 \cdot (-2))}_{\ ext{k-component}\n$$", "This vector lives in 3D space, labeled by the unit vectors $\mathbf{i}, \mathbf{j}, \mathbf{k}$, corresponding to the x, y, and z axes. The scalar expressions inside parentheses represent each component of the vector.", "---", "### Breaking Down the Components", "#### x-component: $ \mathbf{i}(2 \cdot 4 - (-1) \cdot 1) $\nMultiply and subtract inside the parentheses:\n$ 2 \cdot 4 = 8 $,\n$ (-1) \cdot 1 = -1 $,\n$ 8 - (-1) = 8 + 1 = 9 $.\nSo, the x-component is $ 9\mathbf{i} $.", "#### y-component: $ -\mathbf{j}(1 \cdot 4 - (-1)(-2)) $\nEvaluate inside:\n$ 1 \cdot 4 = 4 $,\n$ (-1)(-2) = 2 $, but since it’s subtracted: $ -(-2) $ gives $ - (2) = -2 $,\n$ 4 - 2 = 2 $.\nSo, inside the parentheses is $ 2 $.\nNow apply the negative sign:\n$ - \mathbf{j}(2) = -2\mathbf{j} $.\nThe y-component is $ -2\mathbf{j} $.", "#### z-component: $ \mathbf{k}(1 \cdot 1 - 2 \cdot (-2)) $\nEvaluate inside:\n$ 1 \cdot 1 = 1 $,\n$ 2 \cdot (-2) = -4 $, but with a negative sign: $ - (2 \cdot -2) = -(-4) = +4 $.\nSo, inside: $ 1 + 4 = 5 $.\nThus, the z-component is $ 5\mathbf{k} $.", "---", "### Final Simplified Vector", "Putting it all together:\n$$\n\mathbf{v} = 9\mathbf{i} - 2\mathbf{j} + 5\mathbf{k}\n$$", "Or, in component form:\n$$\n\boxed{(9, -2, 5)}\n$$", "---", "### Why This Matters: Applications in 3D Geometry and Physics", "Vectors like this form the backbone of coordinate geometry, vector calculus, and physics simulations. The computation mirrors operations such as dot products, cross products, and vector projections — all critical in fields like engineering, computer graphics, and mechanical physics.", "For instance, computing forces, velocities, or electric fields often requires breaking down quantities into $\mathbf{i}, \mathbf{j}, \mathbf{k}$ components — a skill directly demonstrated here.", "---", "### Step-by-Step Summary:", "| Step | Operation | Result |\n|-------|--------------------------------------------------|--------------------------|\n| 1 | $2 \cdot 4 - (-1) \cdot 1$ | $8 + 1 = 9$ |\n| 2 | $1 \cdot 4 - (-1)(-2)$ with minus outside | $4 - 2 = 2$, with $-$: $-2$ |\n| 3 | $1 \cdot 1 - 2 \cdot (-2)$ | $1 + 4 = 5$ |\n| 4 | Final vector form in unit vectors | $ \mathbf{i}(9) + \mathbf{j}(-2) + \mathbf{k}(5) $ |\n| 5 | Component form | $ (9, -2, 5) $ |", "---", "### Conclusion", "The expression\n$$\n\mathbf{i}(2 \cdot 4 - (-1) \cdot 1) - \mathbf{j}(1 \cdot 4 - (-1)(-2)) + \mathbf{k}(1 \cdot 1 - 2 \cdot (-2))\n$$\nserves as a clear example of vector construction in 3D space, blending arithmetic with geometric interpretation. Simplifying it yields the vector $ \mathbf{v} = 9\mathbf{i} - 2\mathbf{j} + 5\mathbf{k} $, a concise format useful in diverse applications from scientific computing to game development.", "Understanding such algebraic breakdown not only aids in solving equations but also deepens your intuition for vector spaces — a rewarding step toward mastering linear algebra and applied mathematics.", "---", "Keywords: vector expression, i-j-k components, vector simplification, 3D math, linear algebra tutorial, mathematics education, vector algebra, physics vectors, computational mathematics", "Meta Description:\nUnravel the simplified vector $ \mathbf{i}(2 \cdot 4 - (-1) \cdot 1) - \mathbf{j}(1 \cdot 4 - (-1)(-2)) + \mathbf{k}(1 \cdot 1 - 2 \cdot (-2)) $ into its final form $ (9, -2, 5) $. Learn step-by-step arithmetic and applications in 3D geometry and physics."]