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Question
Find the direction cosines and direction angles of the vector `2hat"i" + hat"j" + 2hat"k"`
Solution
Let `bar"a" = 2hat"i" + hat"j" + 2hat"k"`
`|bar"a"| = sqrt(2^2 + 1^2 + 2^2) = sqrt(4 + 1 + 4) = sqrt9 = 3`
∴ unit vector along bar"a"
`= hat"a" = bar"a"/|bar"a"| = (2hat"i" + hat"j" + 2hat"k")/3 = 2/3hat"i" + 1/3hat"j" + 2/3hat"k"`
∴ its direction cosines are `2/3,1/3,2/3.`
If α, β, γ are the direction angles, then cos α = `2/3`, cos β = `1/3,` cos γ = `2/3`
∴ α = `"cos"^-1(2/3), beta = "cos"^-1(1/3), gamma = "cos"^-1(2/3)`
Hence, direction cosines are `2/3,1/3,2/3` and direction angles are `"cos"^-1(2/3), beta = "cos"^-1(1/3), gamma = "cos"^-1(2/3)`
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