If-a-variable-line-in-two-adjacent-positions-has-direction-cosines-l-m-n-and-l-l-m-m-n-n-show-that-the-small-angle-b-w-the-two-positions-is-given-by-2-l-2-m-2-n-2-

Question Number 75445 by vishalbhardwaj last updated on 11/Dec/19

If a variable line in two adjacent

positions has direction cosines

l, m, n and l + δ l, m + δ m, n + δ n

, show that the small angle δ θ

b / w the two positions is given by

δ θ^{2} = δ l^{2} + δ m^{2} + δ n^{2} ? ?

Answered by mr W last updated on 11/Dec/19

x_{1} = l

y_{1} = m

z_{1} = n

x_{2} = l + δ l

y_{2} = m + δ m

z_{2} = n + δ n

{(Δ l)}^{2} = {(Δ x)}^{2} + {(Δ y)}^{2} + {(Δ z)}^{2}

= {(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2} + {(z_{2} - z_{2})}^{2}

= {(δ l)}^{2} + {(δ m)}^{2} + {(δ n)}^{2} \dots (i)

{(Δ l)}^{2} = 1^{2} + 1^{2} - 2 \times 1 \times 1 \times \cos (δ θ)

= 2 - 2 [1 - 2 \sin^{2} (\frac{δ θ}{2})]

= 4 {[\sin \frac{δ θ}{2}]}^{2}

\approx 4 {[\frac{δ θ}{2}]}^{2}

= {(δ θ)}^{2} \dots (i i)

\Rightarrow {(δ θ)}^{2} = {(δ l)}^{2} + {(δ m)}^{2} + {(δ n)}^{2}

Commented by mr W last updated on 11/Dec/19

Commented by mr W last updated on 11/Dec/19

P_{1} (x_{1}, y_{1}, z_{1})

P_{2} (x_{2}, y_{2}, z_{2})

l e t {O P}_{1} = {O P}_{2} = 1

x_{1} = 1 \cos α = l

y_{1} = 1 \cos β = m

z_{1} = 1 \cos γ = n

s i m i l a r l y

x_{2} = l + δ l

y_{2} = m + δ m

z_{2} = n + δ n

l e t Δ l = P_{1} P_{2}

Δ l = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2} + {(z_{2} - z_{1})}^{2}}

= \sqrt{{(δ l)}^{2} + {(δ m)}^{2} + {(δ n)}^{2}}

\Rightarrow {(Δ l)}^{2} = {(δ l)}^{2} + {(δ m)}^{2} + {(δ n)}^{2}

o n t h e o t h e r s i d e,

{(P_{1} P_{2})}^{2} = {({O P}_{1})}^{2} + {({O P}_{2})}^{2} - 2 ({O P}_{1}) ({O P}_{2}) \cos ∠ P_{1} {O P}_{2}

i . e .

{(Δ l)}^{2} = 1^{2} + 1^{2} - 2 \times 1 \times 1 \cos (δ θ)

= 2 - 2 \cos (δ θ)

= 2 - 2 (1 - 2 \sin^{2} \frac{δ θ}{2})

= 4 {(\sin \frac{δ θ}{2})}^{2}

\approx 4 {(\frac{δ θ}{2})}^{2} s i n c e δ θ i s s m a l l, \sin \frac{δ θ}{2} \approx \frac{δ θ}{2}

= {(δ θ)}^{2}

\Rightarrow {(δ θ)}^{2} = {(δ l)}^{2} + {(δ m)}^{2} + {(δ n)}^{2}

Commented by vishalbhardwaj last updated on 11/Dec/19

sir please explain in detail

Commented by peter frank last updated on 11/Dec/19

t h a n k y o u

Commented by vishalbhardwaj last updated on 11/Dec/19

but sir it is not given that the

length of line is unit

Commented by vishalbhardwaj last updated on 12/Dec/19

And sir this is a particular case

because it is not given that

lines are started from origin,

Infact they can be started from

any point in space except origin

, Please give that general case

to understand .

Commented by mr W last updated on 12/Dec/19

s i n c e w e o n l y n e e d t o f i n d t h e (s m a l l)

c h a n g e o f t h e d i r e c t i o n o f t h e l i n e,

w e c a n c o n s i d e r t h e c h a n g e o f a

u n i t l e n g t h s e g m e n t o f i t a n d w e

c a n t r a n s l a t e t h e s e g m e n t s u c h

t h a t o n e e n d o f t h e s e g m e n t i s a t

t h e s a m e p o i n t . t h e t r a n s l a t i o n

{d o e s n}^{'} t a f f e c t t h e d i r e c t i o n o f t h e

l i n e .

Commented by mr W last updated on 12/Dec/19

Commented by vishalbhardwaj last updated on 12/Dec/19

please sir if it is possible to you

to use general case please post

it

Commented by mr W last updated on 12/Dec/19

i h a v e p o s t e d w h a t i t h i n k i s t h e

c o r r e c t a n s w e r t o t h e q u e s t i o n s i r .

Answered by vishalbhardwaj last updated on 12/Dec/19

Answered by vishalbhardwaj last updated on 12/Dec/19