Question

If a line has direction ratios 2: 1: 2 then what are its direction Cousins

Answer 1

Direction cosines are 2/3,1/3,2/3.

• The direction ratios are given as 2 : 1 : 2.
• The direction ratios will be 2x,x and 2x. (As the direction ratios are given in ratio form , x is the common multiple)
• Now let us consider the direction ratios as a,b,c.
• Then the direction cosines are determined as $\frac{a}{\sqrt{a^{2}+b^{2}+c^{2} } }$,$\frac{b}{\sqrt{a^{2}+b^{2}+c^{2} } }$ and $\frac{c}{\sqrt{a^{2}+b^{2}+c^{2} } }$.
• Determining the direction cosines we get,

$\frac{2x}{\sqrt{{2x}^{2}+x^{2}+{2x}^{2} } }$ = $\frac{2x}{\sqrt{4x^{2}+x^{2}+4x^2 } }$ = $\frac{2x}{\sqrt{9x^2 }$ = $\frac{2}{3}$

Similarly,

$\frac{x}{\sqrt{{2x}^{2}+x^{2}+{2x}^{2} } }$ $\frac{x}{\sqrt{4x^{2}+x^{2}+4x^2 } }$ = $\frac{x}{\sqrt{9x^2 }$ = $\frac{1}{3}$

$\frac{2x}{\sqrt{{2x}^{2}+x^{2}+{2x}^{2} } }$ $\frac{2x}{\sqrt{4x^{2}+x^{2}+4x^2 } }$ = $\frac{2x}{\sqrt{9x^2 }$ = $\frac{2}{3}$

• Therefore the direction cosines are 2/3,1/3.2/3.