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
Made by
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.
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