Question

For direction ratios \[(a,b,c)\], which set can represent the corresponding direction cosines when the direction of the line is chosen?

Options

• \[l=\sqrt{a^2+b^2+c^2},\quad m=\sqrt{a^2+b^2+c^2},\quad n=\sqrt{a^2+b^2+c^2}\]

• \[l=\frac{1}{a},\quad m=\frac{1}{b},\quad n=\frac{1}{c}\]

• \[l=\frac{a}{a^2+b^2+c^2},\quad m=\frac{b}{a^2+b^2+c^2},\quad n=\frac{c}{a^2+b^2+c^2}\]

• \[l=\frac{a}{\sqrt{a^2+b^2+c^2}},\quad m=\frac{b}{\sqrt{a^2+b^2+c^2}},\quad n=\frac{c}{\sqrt{a^2+b^2+c^2}}\]

Answer 1

Direction cosines are obtained by dividing each direction ratio by the magnitude \[\sqrt{a^2+b^2+c^2}\]. This normalization makes \[l^2+m^2+n^2=1\].