The equation of a line is $fraction numerator 2 straight x minus 5 over denominator 4 end fraction space equals space fraction numerator straight y plus 4 over denominator 3 end fraction space equals fraction numerator 6 minus straight z over denominator 6 end fraction$ . Find the direction cosines of a line parallel to this line.

Solution

The equation of given line is

fraction numerator 2 straight x minus 5 over denominator 4 end fraction space equals fraction numerator straight y plus 4 over denominator 3 end fraction space equals fraction numerator 6 minus straight z over denominator 6 end fraction

fraction numerator straight x minus begin display style 5 over 2 end style over denominator 2 end fraction space equals space fraction numerator straight y plus 4 over denominator 3 end fraction space equals space fraction numerator straight z minus 6 over denominator negative 6 end fraction

Its direction ratios are 2, 3, -6
The direction ratios of a line parallel to given line are 2, 3. - 6

therefore

direction cosines are

fraction numerator 2 over denominator square root of 4 plus 9 plus 36 end root end fraction comma space fraction numerator 3 over denominator square root of 4 plus 9 plus 36 end root end fraction comma space fraction numerator negative 6 over denominator square root of 4 plus 9 plus 36 end root end fraction

i.e.

fraction numerator 2 over denominator square root of 49 end fraction comma space fraction numerator 3 over denominator square root of 49 end fraction comma space fraction numerator negative 6 over denominator square root of 49 end fraction

i.e.

2 over 7 comma space 3 over 7 comma space minus 6 over 7

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