If the direction cosines of a variable line in two adjacent positions be l, m, n

Question

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 :

Options:

A . θ=a+b+c

B . θ2=a2+b2+c2

C . |θ|=|a|+|b|+|c|

D . θ3=a3+b3+c3

Answer: Option B
:
B

l^{2} + m^{2} + n^{2} = 1, (l + a)^{2} + (m + b)^{2} + (n + c)^{2} = 1 \Rightarrow a l + b m + c n = - \frac{1}{2} (a^{2} + b^{2} + c^{2})

If The Direction Cosines Of A Variable Line In Two Adjacent ...

c o s θ = (a + l) l + m (b + m) + n (c + n)

= 1 + al + bm + cn

\Rightarrow a^{2} + b^{2} + c^{2} = 2 (1 - c o s θ) = 4 s i n^{2} (\frac{θ}{2})

Since

θ

is small,

s i n (\frac{θ}{2}) \approx \frac{θ}{2}

∴ θ^{2} = a^{2} + b^{2} + c^{2} [∵ s i n^{2} (\frac{θ}{2}) \approx \frac{θ^{2}}{4}]

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