#### 8Question: Given three vertices of a cube at \((1, 2, 3)\), \((1, 5, 3)\), and \((4, 2, 3)\), find the coordinates of the fourth vertex that is diagonally opposite to \((1, 2, 3)\) on the same face, assuming all coordinates are integers.

["Finding the Diagonally Opposite Vertex of a Cube: A Step-by-Step Guide Given Three Vertices", "When working with geometric shapes like cubes, determining missing coordinates can be challenging—but with careful analysis, it becomes straightforward. This article solves a practical problem: Given three vertices of a cube at ((1, 2, 3)), ((1, 5, 3)), and ((4, 2, 3)), find the integer coordinates of the fourth vertex that lies diagonally opposite to ((1, 2, 3)) on the same face of the cube.", "---", "### Understanding the Cube Geometry", "A cube is composed of 8 equal-length edges and 6 square faces. Diagonally opposite on a face means lying at the farthest corner from a given vertex, across the square face—connected via face diagonal, not through the cube’s 3D diagonal. This vertex shares two face edges with the given points.", "---", "### Step 1: Analyze Given Points and Face Alignment", "The three points provided are:\n- (A = (1, 2, 3))\n- (B = (1, 5, 3))\n- (C = (4, 2, 3))", "All three points have the same (z)-coordinate: 3, meaning they lie on the same horizontal plane ((z = 3)). This confirms they belong to a single face of the cube — specifically, the bottom face if the cube is aligned with the coordinate axes.", "Now compare coordinates to deduce edge lengths:", "- From (A) to (B):\n (x) fixed at 1, (y) goes from 2 to 5 → difference: (5 - 2 = 3)\n (z) is constant → edge in (y)-direction, length = 3.", "- From (A) to (C):\n (y) fixed at 2, (x) goes from 1 to 4 → difference: (4 - 1 = 3)\n (z) constant → edge in (x)-direction, length = 3.", "Since both vectors ((0, 3, 0)) and ((3, 0, 0)) represent edges of the cube, and both have length 3, the face lies on the plane (z = 3), aligned with the (x)-(y) plane.", "Thus, the three points form a right corner of a square face with edge length 3.", "---", "### Step 2: Identify Missing Corner on This Face", "On a square face with vertices ((1,2,3)), ((1,5,3)), and ((4,2,3)), the missing vertex must complete the rectangle (which is a square here) by connecting the unshared edges.", "These are the three known corners:\n- (A = (1,2,3))\n- (B = (1,5,3)) (same (x), (z); differing (y))\n- (C = (4,2,3)) (same (y), (z); differing (x))", "The only missing corner opposite (A) is the one that completes the opposite corner from (A) — meaning it shares the unconnected directions:\n- Along the (x)-axis from (C): (x = 4)\n- Along the (y)-axis from (B): (y = 5)\n- Constant (z = 3)", "Thus, the fourth vertex is:\n[\n(4, 5, 3)\n]", "---", "### Step 3: Confirm Diagonal Opposition on the Face", "Let’s verify this point is diagonally opposite to (A = (1,2,3)) across the face:", "- Vector from (A) to (B): ((0, 3, 0))\n- Vector from (A) to (C): ((3, 0, 0))\n- Vector from (A) to ((4,5,3)): ((3, 3, 0)) → sum of diagonal vectors.", "Indeed, the diagonal from (A = (1,2,3)) goes through ((4,2,3)) and ((1,5,3)), and to complete the square, the missing diagonally opposite vertex must be the sum of the displacement:\n[\nA + (B - A) + (C - A) = (1,2,3) + (0,3,0) + (3,0,0) = (4,5,3)\n]", "Alternatively, since diagonal of a square connects opposite corners, and all edges are axis-aligned (cube edges), the coordinates differ independently in (x) and (y):\n- (x): from 1 to 4 → full extension along (x)\n- (y): from 2 to 5 → full extension along (y)\n- (z): remains 3 (same face)", "Hence, ((4,5,3)) is the only missing corner forming a square with the three given points.", "---", "### Step 4: Why Diagonal Opposite On the Face, Not Through the Cube?", "While a cube has one face diagonal (from (A) to ((4,5,3))) that spans the face, the question specifies the fourth vertex on the same face diagonally opposite to (A). This is not through the interior of the cube but within the square face—so vector addition in the plane suffices.", "---", "### Final Answer", "The coordinates of the fourth vertex diagonally opposite to ((1, 2, 3)) on the shared face are:", "((4, 5, 3))", "---", "### SEO Keywords & Optimization:", "- primary keywords: “finding diagonally opposite vertex on cube face”, “cube vertex given three coordinates”, “coordinates of fourth cube corner”, “3D geometry cube face diagonal”\n- long-tail keywords: “cube diagonally opposite vertex on the same face given three vertices”, “integer coordinates cube face diagonal”, “how to find missing cube vertex on plane”\n- structure:\n - Clear heading with question\n - Step-by-step logical breakdown\n - Example with coordinates and vectors\n - Confirmation via diagonal logic\n - Emphasis on face-only diagonal, not 3D space", "This structured approach improves readability, supports SEO through semantic keyword integration, and ensures users understand both geometry and application—ideal for technical quizzes, geometry learners, and coding problem solvers."]