Question

If the direction cosines of a line is <2/7,3/7,k/7>then find the value of k?

Answer 1

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

Given that

Direction cosines of the line is $\bigg(\dfrac{2}{7}, \dfrac{3}{7},\dfrac{k}{7} \bigg )$

We know,

If the direction cosines of the line is (l, m, n) then

$\boxed{ \: {l}^{2} + {m}^{2} + {n}^{2} \: = \: 1 \: }$

Here,

$\: l \: = \: \dfrac{2}{7}$

$\: m \: = \: \dfrac{3}{7}$

$\: n \: = \: \dfrac{k}{7}$

So, on substituting the values, we get

$\rm \: {\bigg(\dfrac{2}{7} \bigg) }^{2} + {\bigg(\dfrac{3}{7} \bigg) }^{2} + {\bigg(\dfrac{k}{7} \bigg) }^{2} = 1$

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

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

$\rm \: \dfrac{13 + {k}^{2} }{49} = 1$

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

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

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

$\rm\implies \:\boxed{ \rm{ \:k \: = \: \pm \: 6 \: \: }}$

$\rule{190pt}{2pt}$

Additional Information :-

1. The direction cosines of x - axis is (1, 0, 0)

2. The direction cosines of y - axis is (0, 1, 0)

3. The direction cosines of z - axis is (0, 0, 1)