3D Geometry
Vectors and Quadrilaterals
Grade 12
Question:
<p>The points \((5, -4, 2)\), \((4, -3, 1)\), \((7, -6, 4)\) and \((8, -7, 5)\) are the vertices of:</p>
<p>(a) a rectangle</p>
<p>(b) a square</p>
<p>(c) a parallelogram</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Use position vectors to find the sides of the quadrilateral. Check if opposite sides are equal vectors (which indicates a parallelogram) and if consecutive sides are perpendicular (which would indicate a rectangle).
Solution: Let \(A(5, -4, 2)\), \(B(4, -3, 1)\), \(C(7, -6, 4)\) and \(D(8, -7, 5)\). Compute the vectors: \[\vec{AB} = -\hat{i} + \hat{j} - \hat{k}\] \[\vec{BC} = 3\hat{i} - 3\hat{j} + 3\hat{k}\] To determine the type of quadrilateral, check if opposite sides are parallel and equal. Since the corresponding components show that opposite sides are parallel, this is a parallelogram. ∴ Answer is (c).
Correct Answer: C