This question is similar to CBSE Class 12 Sample Paper for 2021 Boards

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https://www.teachoo.com/12406/3415/Question-26/category/CBSE-Class-12-Sample-Paper-for-2021-Boards/

Question 25

The two co-initial adjacent sides of a parallelogram are 2ı ˆ-4ȷ ˆ-5k ˆ and 2ı ˆ+2ȷ ˆ+3k ˆ. Find its diagonals and use them to find the area of the parallelogram.

 

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Question 25 The two co-initial adjacent sides of a parallelogram are 2ı ˆ−4ȷ ˆ−5𝑘 ˆ and 2ı ˆ+2ȷ ˆ+3𝑘 ˆ. Find its diagonals and use them to find the area of the parallelogram.Let ABCD be the parallelogram Where 𝑎 ⃗ = 2𝚤 ˆ−4𝚥 ˆ−5𝑘 ˆ 𝑏 ⃗ = 2𝚤 ˆ+2𝚥 ˆ+3𝑘 ˆ We need to find diagonals (𝑑_1 ) ⃗ and (𝑑_2 ) ⃗ Finding diagonal (𝒅_𝟏 ) ⃗ Now, (𝒅_𝟏 ) ⃗=𝒂 ⃗ + 𝒃 ⃗ = (2𝚤 ˆ−4𝚥 ˆ−5𝑘 ˆ) + (2𝚤 ˆ+2𝚥 ˆ+3𝑘 ˆ) = 𝟒𝚤 ˆ−𝟐𝚥 ˆ−𝟐𝒌 ˆ Finding diagonal (𝒅_𝟐 ) ⃗ Now, (𝒅_𝟐 ) ⃗=𝒂 ⃗ − 𝒃 ⃗ = (2𝚤 ˆ−4𝚥 ˆ−5𝑘 ˆ) − (2𝚤 ˆ+2𝚥 ˆ+3𝑘 ˆ) = 2𝚤 ˆ−4𝚥 ˆ−5𝑘 ˆ − 2𝚤 ˆ−2𝚥 ˆ−3𝑘 ˆ = −𝟔𝚥 ˆ−𝟖𝒌 ˆ Now, We need to find area of parallelogram using diagonals Area of parallelogram = 𝟏/𝟐 |(𝒅_𝟏 ) ⃗" × " (𝒅_𝟐 ) ⃗ | Finding (𝒅_𝟏 ) ⃗" × " (𝒅_𝟐 ) ⃗ (𝒅_𝟏 ) ⃗" × " (𝒅_𝟐 ) ⃗ = |■8(𝑖 ̂&𝑗 ̂&𝑘 ̂@4&−2&−2@0&−6&−8)| = 𝑖 ̂ ((−2) × (−8) − (−6) × (−2)) − 𝑗 ̂ (4 × (−8) − 0 × (−2)) + 𝑘 ̂ (4 × (−6) − 0 × (−2)) = 𝑖 ̂ (16 − 12) − 𝑗 ̂ (−32 − 0) + 𝑘 ̂ (−24 − 0) = 4𝑖 ̂ + 32𝑗 ̂ − 24𝑘 ̂ = 4(𝒊 ̂ + 8𝒋 ̂ − 6𝒌 ̂) Now, Area of Parallelogram ABCD = 𝟏/𝟐 |(𝒅_𝟏 ) ⃗" × " (𝒅_𝟐 ) ⃗ | = 1/2 × 4√(12+82+(−6)2) = 2√(1+64+36) = 𝟐√𝟏𝟎𝟏 square units

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.