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Class 12 Vector Algebra · Long Answer3 Markscross_product_area

Class 12 Maths · Vector Algebra · Long Answer #66

Adjacent sides of a parallelogram are u=(2,−4,−5), v=(2,2,3). Find unit vectors along both diagonals and calculate the area using the diagonals.
Parallelogram and its diagonal segments
Parallelogram and its diagonal segments

✅ Step-by-Step Solution

Final Answer: Unit directions ±(4,−2,−2)/(2√6), ±(0,6,8)/10; area=2√101
  1. The diagonals are d₁=u+v=(4,−2,−2), d₂=v−u=(0,6,8).
  2. Their norms are 2√6 and 10. Thus their unit directions are ±d₁/(2√6) and ±d₂/10.
  3. For diagonals, area=½|d₁×d₂|. Compute d₁×d₂=(−4,−32,24), of squared norm 1616.
  4. Therefore area=½√1616=2√101 square units.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] The diagonals are d₁=u+v=(4,−2,−2), d₂=v−u=(0,6,8).
  • [1 mark] Their norms are 2√6 and 10. Thus their unit directions are ±d₁/(2√6) and ±d₂/10.
  • [1 mark] For diagonals, area=½|d₁×d₂|. Compute d₁×d₂=(−4,−32,24), of squared norm 1616.
  • [0.5 marks] Therefore area=½√1616=2√101 square units.

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