Solution: The given points lie on the plane \(z = 3\). The vector from \((1, 2, 3)\) to \((1, 5, 3)\) is \((0, 3, 0)\), and the vector to \((4, 2, 3)\) is \((3, 0, 0)\). The fourth vertex on the same face is obtained by adding these vectors: \((1 + 3, 2 + 3, 3) = (4, 5, 3)\).
["Solution: Determining the Fourth Vertex on the Plane ( z = 3 ) Using Vector Addition", "In 3D geometry, understanding how points and vectors interact on a plane is essential for solving spatial problems—especially in fields like computer graphics, engineering, and spatial analysis. This article explains a key concept using a real-world geometric scenario: finding a fourth vertex on the plane ( z = 3 ) by combining vector operations.", "---", "### The Problem Setup", "We are working within the plane defined by ( z = 3 )—a flat, horizontal surface at constant height. Given three points with coordinates:", "- ( A = (1, 2, 3) )\n- ( B = (1, 5, 3) )\n- ( C = (4, 2, 3) )", "We’re told:", "- The vector ( \vec{AB} = B - A = (0, 3, 0) )\n- The vector ( \vec{AC} = C - A = (3, 0, 0) )\n- We seek the fourth vertex ( D ) on the same face (plane) such that it lies at the sum of these two displacement vectors added starting from point ( A )", "---", "### Step-by-Step Explanation", "#### Step 1: Confirm All Points Lie on the Plane\nAll given points ( (1,2,3) ), ( (1,5,3) ), and ( (4,2,3) ) have ( z = 3 ), confirming they lie on the plane ( z = 3 ).", "#### Step 2: Compute Relevant Vectors\nThe vector ( \vec{AB} = (0, 3, 0) ) defines movement 3 units in the positive ( y )-direction.\nThe vector ( \vec{AC} = (3, 0, 0) ) defines movement 3 units in the positive ( x )-direction.", "#### Step 3: Add the Vectors to Find the Missing Vertex\nTo locate the fourth vertex ( D ) on the same flat face and completing the parallelogram-like face formed by these points, we add the two vectors starting from point ( A ):\n[\n\vec{AD} = \vec{AB} + \vec{AC} = (0, 3, 0) + (3, 0, 0) = (3, 3, 0)\n]", "Then, compute the coordinates of ( D ) by adding this vector to ( A ):\n[\nD = A + \vec{AD} = (1, 2, 3) + (3, 3, 0) = (4, 5, 3)\n]", "---", "### Why This Works\nAdding vectors representing displacements from a common base point (here, ( A )) produces a new point completing the geometric figure in the plane. Since all points lie in ( z = 3 ), the result naturally preserves ( z = 3 ).", "---", "### Conclusion: The Missing Vertex\nThe fourth vertex on the plane bounded by these points is:\n[\n\boxed{(4, 5, 3)}\n]", "This method demonstrates how vector addition aids in reconstructing spatial configurations—critical for architectural modeling, robotics navigation, and 3D rendering. Understanding such spatial relationships empowers accurate problem-solving in geometry and beyond.", "---", "Keywords:\nplane ( z = 3 ), four vectors in 3D geometry, vector addition, parallelogram on a plane, coordinate geometry, spatial analysis, coordinate geometry solution, vector addition in 3D space."]
