Question

If the dcs of a line be < 2/7,3/7,k/7> what is the value of k?​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

$\rm :\longmapsto\:d.c's \: of \: a \: line \: are \: < \dfrac{2}{7} , \: \dfrac{3}{7}, \: \dfrac{k}{7} >$

We know,

If < l, m, n > represents the direction cosines of line then

$\red{\rm :\longmapsto\:\boxed{\tt{ {l}^{2} + {m}^{2} + {n}^{2} = 1 \: }}}$

So, here

$\red{\rm :\longmapsto\:l \: = \: \dfrac{2}{7}}$

$\red{\rm :\longmapsto\:m \: = \: \dfrac{3}{7}}$

$\red{\rm :\longmapsto\:n \: = \: \dfrac{k}{7}}$

So,

$\red{\rm :\longmapsto\: {\bigg[\dfrac{2}{7} \bigg]}^{2} + {\bigg[\dfrac{3}{7} \bigg]}^{2} + {\bigg[\dfrac{k}{7} \bigg]}^{2} = 1}$

$\red{\rm :\longmapsto\:\dfrac{4}{49} + \dfrac{9}{49} + \dfrac{ {k}^{2} }{49} = 1}$

$\red{\rm :\longmapsto\:\dfrac{4 + 9 + {k}^{2} }{49} = 1}$

$\red{\rm :\longmapsto\:13 + {k}^{2} = 49}$

$\red{\rm :\longmapsto\:{k}^{2} = 49 - 13}$

$\red{\rm :\longmapsto\:{k}^{2} = 36}$

$\red{\rm :\longmapsto\:k = \pm \: 6}$

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Additional Information

1. Direction cosines of a line is defined as cosine of the angle which a line makes with respective axis. If line makes an angle a, b and c with respective axis, then direction cosine of the line is denoted by l, m, n and is given by

$\green{\rm :\longmapsto\:l = cosa}$

$\green{\rm :\longmapsto\:m = cosb}$

$\green{\rm :\longmapsto\:n = cosc}$

$\blue{\rm :\longmapsto\: \: (1) \: \: {cos}^{2}a + {cos}^{2}b + {cos}^{2}c = 1}$

$\blue{\rm :\longmapsto\: \: (2) \: \: {sin}^{2}a + {sin}^{2}b + {sin}^{2}c = 2}$