Vector Algebra
Position Vectors and Geometry
Grade 12
Question:
<p>Consider points <span>\(A\)</span>, <span>\(B\)</span>, <span>\(C\)</span> and <span>\(D\)</span> with position vectors <span>\(7\hat{i} - 4\hat{j} + 7\hat{k}\)</span>, <span>\(\hat{i} - 6\hat{j} + 10\hat{k}\)</span>, <span>\(-\hat{i} - 3\hat{j} + 4\hat{k}\)</span> and <span>\(5\hat{i} - \hat{j} + \hat{k}\)</span>, respectively. Then <span>\(ABCD\)</span> is a</p>
<p>square.</p>
<p>rhombus.</p>
<p>rectangle.</p>
<p>parallelogram but not a rhombus.</p>
Step-by-Step Solution
Key Concept: A quadrilateral ABCD is a parallelogram if and only if its diagonals bisect each other, meaning the midpoint of AC equals the midpoint of BD. Calculate both midpoints and compare.
Step 1: Find midpoint of diagonal AC. Midpoint of AC = (A + C)/2 = [(7î - 4ĵ + 7k̂) + (-î - 3ĵ + 4k̂)]/2 = (6î - 7ĵ + 11k̂)/2 = 3î - 3.5ĵ + 5.5k̂ Step 2: Find midpoint of diagonal BD. Midpoint of BD = (B + D)/2 = [(î - 6ĵ + 10k̂) + (5î - ĵ + k̂)]/2 = (6î - 7ĵ + 11k̂)/2 = 3î - 3.5ĵ + 5.5k̂ Step 3: Compare midpoints. Since midpoint of AC = midpoint of BD, the diagonals bisect each other. Step 4: Verify it's not a rectangle (optional). Check if |AB| = |BC| (which would indicate a square) or if diagonals are equal in length. Since the diagonals bisect each other, ABCD is at minimum a parallelogram . ∴ Answer: B (Parallelogram)
Correct Answer: B