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Show that the lines x- 5/ 7 = y+ 2/- 3 = z/1 and x/ 1 = y/2 = z/3 are perpendicular to each other.

13. Show that the lines \frac{x-5}{7}=\frac{y+2}{-3}=\frac{z}{1} and \frac{x}{1}=\frac{y}{2}=\frac{z}{3} are perpendicular to each other.

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First, we have to write the given equation of lines in the standard form;

 \frac{x-5}{7}=\frac{y+2}{-5}=\frac{z}{1} and \frac{x}{1}=\frac{y}{2}=\frac{z}{3}

Then we have the direction ratios of the above lines as;

7,\ -5,\ 1   and   1,\ 2,\ 3   respectively..

Two lines with direction ratios a_{1},b_{1},c_{1} and a_{2},b_{2},c_{2}  are perpendicular to each other if,  a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2}= 0

\therefore 7(1) + (-5)(2)+1(3) = 7-10+3 = 0

Therefore the two lines are perpendicular to each other.

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