Misc 11 - Show direction cosines of a vector equally inclined - Direction cosines and ratios


  1. Chapter 10 Class 12 Vector Algebra
  2. Serial order wise
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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﷯ = 𝑎 𝑖﷯ + b 𝑗﷯ + c 𝑘﷯ 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﷯ = 𝑎 𝑖﷯ + 𝑎 𝑗﷯ + 𝑎 𝑘﷯ 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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