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Misc 11 - Show direction cosines of a vector equally inclined

Misc 11 - Chapter 10 Class 12 Vector Algebra - Part 2

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Misc 11 Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are 1/√3, 1/√3, 1/√3 . Let the required vector be 𝒓 ⃗ = 𝒂𝒊 ̂ + b𝒋 ̂ + c𝒌 ̂ Directions ratios are 𝑎, 𝑏, and 𝑐. Since the vector is equally inclined to axes OX, OY and OZ, thus the direction cosines are equal. 𝑎/(𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 𝑜𝑓 𝑟 ⃗ ) = 𝑏/(𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 𝑜𝑓 𝑟 ⃗ ) = 𝑐/(𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 𝑜𝑓 𝑟 ⃗ ) 𝑎 = b = c ∴ The vector is 𝑟 ⃗ = 𝑎𝑖 ̂ + 𝑎𝑗 ̂ + 𝑎𝑘 ̂ Magnitude of 𝑟 ⃗ = √(𝑎2+𝑎2+𝑎2) |𝑟 ⃗ | = √3𝑎2 |𝑟 ⃗ | = √3 𝑎 Direction cosines are (𝑎/(√3 𝑎),𝑏/(√3 𝑎),𝑐/(√3 𝑎)) = (𝑎/(√3 𝑎),𝑎/(√3 𝑎),𝑎/(√3 𝑎)) = (𝟏/√𝟑,𝟏/√𝟑,𝟏/√𝟑) Hence proved

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.