# The direction cosines of a variable I in 2 adj. positions are LMN , L+(DELTA)N , N+(DELTA )M ,AND M +(DELTA)L. Show that 4 small angle delta theta, b/w 2 positions is given by : [delta theta]^2 = {(DELTA)L}^2+[(DELTA)M]^2+[(DELTA)N]^2

Aman Bansal
592 Points
12 years ago

Dear Jasmine,

Using the property,

If v is a vector

${\mathbf v}= v_1 \boldsymbol{\hat{x}} + v_2 \boldsymbol{\hat{y}} + v_3 \boldsymbol{\hat{z}}$

where $\boldsymbol{\hat{x}}, \boldsymbol{\hat{y}}, \boldsymbol{\hat{z}}$ is a basis. Then the direction cosines are

\begin{align} \alpha & = \cos a = \frac{{\mathbf v} \cdot \boldsymbol{\hat{x}} }{ \left \Vert {\mathbf v} \right \Vert } & = \frac{v_1}{\sqrt{v_1^2 + v_2^2 + v_3^2}} ,\\ \beta & = \cos b = \frac{{\mathbf v} \cdot \boldsymbol{\hat{y}} }{ \left \Vert {\mathbf v} \right \Vert } & = \frac{v_2}{\sqrt{v_1^2 + v_2^2 + v_3^2}} ,\\ \gamma &= \cos c = \frac{{\mathbf v} \cdot \boldsymbol{\hat{z}} }{ \left \Vert {\mathbf v} \right \Vert } & = \frac{v_3}{\sqrt{v_1^2 + v_2^2 + v_3^2}}. \end{align}

Note that

α2 + β2 + γ2 = 1

And approximation ,

we can get the required answer.

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Aman Bansal