If the direction cosines of a variable line in two adjacent positions be l, m, n and l + a, m + b, n + c and the small angle between the two positions be θ, then :
A
θ=a+b+c
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B
θ2=a2+b2+c2
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C
|θ|=|a|+|b|+|c|
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D
θ3=a3+b3+c3
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Solution
The correct option is Bθ2=a2+b2+c2 l2+m2+n2=1,(l+a)2+(m+b)2+(n+c)2=1⇒al+bm+cn=−12(a2+b2+c2)
cosθ=(a+l)l+m(b+m)+n(c+n)
= 1 + al + bm + cn ⇒a2+b2+c2=2(1−cosθ)=4sin2(θ2)
Since θ is small, sin(θ2)≈θ2 ∴θ2=a2+b2+c2[∵sin2(θ2)≈θ24]