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Misc 10 - Find unit vector parallel to parallelogram diagonal - Vector product - Area

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  1. Chapter 10 Class 12 Vector Algebra
  2. Serial order wise
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Misc 10 The two adjacent sides of a parallelogram are 2 𝑖﷯ − 4 𝑗﷯ + 5 𝑘﷯ and 𝑖﷯ − 2 𝑗﷯ − 3 𝑘﷯ Find the unit vector parallel to its diagonal. Also, find its area. Let 𝑎﷯ and 𝑏﷯ are adjacent side of a parallelogram, where 𝑎﷯ = 2 𝑖﷯ − 4 𝑗﷯ + 5 𝑘﷯ 𝑏﷯ = 𝑖﷯ − 2 𝑗﷯ − 3 𝑘﷯ Let diagonal be 𝑐﷯ Hence, 𝒄﷯ = 𝒂﷯ + 𝒃﷯ = (2 𝑖﷯ − 4 𝑗﷯ + 5 𝑘﷯) + (1 𝑖﷯ − 2 𝑗﷯ − 3 𝑘﷯) = (2 + 1) 𝑖﷯ − (4 + 2) 𝑗﷯ + (5 − 3) 𝑘﷯ = 3 𝒊﷯ − 6 𝒋﷯ + 2 𝒌﷯ Magnitude of 𝑐﷯﷯ = ﷮ 3﷯﷮2﷯+ −6﷯﷮2﷯+ 2﷯﷮2﷯﷯ = ﷮9+36+4﷯ = ﷮49﷯ = 7 Unit vector in direction of 𝑐﷯ = 1﷮𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 𝑜𝑓 𝑐﷯﷯ × 𝑐﷯ 𝒄﷯ = 𝟏﷮𝟕﷯ 3 𝒊﷯ − 6 𝒋﷯ + 2 𝒌﷯﷯ Area of parallelogram = 𝑎﷯× 𝑏﷯﷯ So 𝑎﷯× 𝑏﷯ = 𝑖﷯﷮ 𝑗﷯﷮ 𝑘﷯﷮2﷮−4﷮5﷮1﷮−2﷮−3﷯﷯ = 𝑖﷯ −3×−4﷯−(−2×5)﷯ − 𝑗﷯ 2×−3﷯−(1×5)﷯ + 𝑘﷯ −2×2﷯−(−4×1)﷯ = 𝑖﷯ (12 − (−10) − 𝑗﷯ (−6 −5) + 𝑘﷯ (−4 − (−4)) = 𝑖﷯ (12 + 10) − 𝑗﷯ (−11) + 𝑘﷯ (−4 + 4) = 22 𝑖﷯ + 11 𝑗﷯ + 0 𝑘﷯ = 22 𝑖﷯ + 11 𝑗﷯ So 𝒂﷯× 𝒃﷯ = 22 𝒊﷯ + 11 𝒋﷯ 𝑎﷯× 𝑏﷯﷯ = ﷮ 22﷮2﷯+ 11﷮2﷯﷯ = ﷮ 2﷮2﷯( 11)﷮2﷯+ 11﷮2﷯﷯ = ﷮ 11﷮2﷯( 2﷮2﷯+1)﷯ = 11 ﷮5﷯ Hence, Area of parallelogram = 𝑎﷯× 𝑏﷯﷯= 𝟏𝟏 ﷮𝟓﷯

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