Question

what do you understand by direction of cosines explain it by resolution of vectors in 3D

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Answer 1

Direction Cosines :

Direction cosines are numerical values used to define a vector. These are the cosines of the angle they make with the x,y and z axes.

• l is a direction cosine with x axis
• m is a direction cosine with y axis
• n is a direction cosine with z axis

Example:

Let there be a vector A which makes angle :

$\displaystyle \rm \alpha \: with \: x - axis$

$\displaystyle \rm \beta \: with \: y - axis$

$\displaystyle \rm \gamma \: with \: z - axis$

Then, we define the direction cosines as :

$\displaystyle \rm \bull l = cos (\alpha)$

$\displaystyle \rm \bull m= cos (\beta)$

$\displaystyle \rm \bull n= cos (\gamma)$

Some Properties Of Direction Cosines :

$\displaystyle \rm \bull {l}^{2} + {m}^{2} + {n}^{2} = 1$

$\displaystyle \rm \implies cos {}^{2} ( \alpha) + cos {}^{2} ( \beta) + cos {}^{2} ( \gamma) = 1$

Proof :

$\displaystyle \rm \implies cos {}^{2} ( \alpha) + cos {}^{2} ( \beta) + cos {}^{2} ( \gamma)$

$\displaystyle \rm \implies {\left(\frac{ |\overrightarrow{Ax}| }{ |\overrightarrow{A}| }\right)}^{2} + { \left( \frac{ | \overrightarrow{Ay}| }{ |\overrightarrow{A}| } \right) }^{2} + { \left( \frac{ |\overrightarrow{Az}| }{ |\overrightarrow{A}| } \right) }^{2}$

$\displaystyle \rm \implies \frac{{Ax} {}^{2} }{A {}^{2} } + \frac{{Ay} {}^{2} }{A {}^{2} } + \frac{{Az} {}^{2} }{A {}^{2} }$

$\displaystyle \rm \implies \frac{{Ax} {}^{2} + {Ay} {}^{2} + {Az} {}^{2} }{A {}^{2} }$

$\displaystyle \rm \implies \frac{{A} {}^{2} }{A {}^{2} }$

$\displaystyle \rm \implies {l}^{2} + {m}^{2} + {n}^{2} = 1$

Hence proved