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Example 25 - Find area of a parallelogram whose a = 3i + j + 4k

Example 25 - Chapter 10 Class 12 Vector Algebra - Part 2

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Example 25 Find the area of a parallelogram whose adjacent sides are given by the vectors (π‘Ž ) βƒ— = 3𝑖 Μ‚ + 𝑗 Μ‚ + 4π‘˜ Μ‚ and 𝑏 βƒ— = 𝑖 Μ‚ βˆ’ 𝑗 Μ‚ + π‘˜ Μ‚ Given (𝒂 ) βƒ— = 3𝑖 Μ‚ + 1𝑗 Μ‚ + 4π‘˜ Μ‚ 𝒃 βƒ— = 1𝑖 Μ‚ βˆ’ 1𝑗 Μ‚ + 1k Μ‚ Area of parallelogram ABCD = |𝒂 βƒ— Γ— 𝒃 βƒ— | Now, (𝒂 ) βƒ—Γ— 𝒃 βƒ— = |β– 8(𝑖 Μ‚&𝑗 Μ‚&π‘˜ Μ‚@3&1&4@1&βˆ’1&1)| = 𝑖 Μ‚ (1 Γ— 1 – (βˆ’1) Γ— 4) βˆ’ 𝑗 Μ‚ (3 Γ— 1 – 1 Γ— 4) + π‘˜ Μ‚ (3 Γ— βˆ’1 βˆ’ 1 Γ— 1) = 𝑖 Μ‚(1 βˆ’ (-4)) βˆ’ j Μ‚ (3 βˆ’ 4) + π‘˜ Μ‚ (βˆ’3 βˆ’1) = 𝑖 Μ‚(1 + 4) βˆ’ j Μ‚ (βˆ’1) + π‘˜ Μ‚ (βˆ’4) = 5π’Š Μ‚ + 𝒋 Μ‚ βˆ’ 4π’Œ Μ‚ Magnitude of π‘Ž βƒ— Γ— 𝑏 βƒ— = √(52+1^2+(βˆ’4)2) |𝒂 βƒ— Γ— 𝒃 βƒ— | = √(25+1+16) = βˆšπŸ’πŸ Area of parallelogram ABCD = |π‘Ž βƒ— Γ— 𝑏 βƒ— | = √42 Therefore, the required area is βˆšπŸ’πŸ .

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