#### Question

Find the direction cosines of the line perpendicular to the lines whose direction ratios are -2, 1,-1 and -3, - 4, 1

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#### Solution

Given direction ratios are:

2,1, 1 and 3, 4,1

Let a, b and c be the direction ratios of the line perpendicular to the given lines.

Thus, we have,

-2a + b - c = 0

-3a - 4b+c = 0

Cross multiplying, we get,

`a/(1xx1-(-4)xx(-1))=b/((-3)xx(-1)-(-2)xx1)=c/((-2)xx(-4)-(-3)xx1)`

`a/(1-4)=b/(3+2)=c/(8+3)`

`a/-3=b/5=c/11`

Let us find `sqrt(a^2_b^2+c^2)`

`sqrt(a^2_b^2+c^2)=sqrt((-3)^2_5^2+11^2)`

= `sqrt(9+25+121)=sqrt(155)`

Thus, the direction ratios of the required line are -3,5,11

The direction cosines are: `-3/sqrt155,5/sqrt155,11/sqrt155`

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#### Reference Material

Solution for question: Find the Direction Cosines of the Line Perpendicular to the Lines Whose Direction Ratios Are -2, 1,-1 and -3, - 4, 1 concept: Direction Cosines and Direction Ratios of a Line. For the courses HSC Science (Electronics), HSC Science (Computer Science), HSC Arts, HSC Science (General)