Show that the lines x-5-7-y-2-5-z-1 and x-1-y-2-z-3 perpendicular to each other-

The equations of the given lines are x 5/7=y+2/ 5=z/1 and x/1=y/2=z/3 direction ratios of the given lines are respectively (7, 5,1) and (1, 2, 3). Two lines w ...

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Standard XII

Mathematics

Distance between Two Parallel Planes

Show that the...

Question

Show that the lines $\frac{x - 5}{7} = \frac{y + 2}{- 5} = \frac{z}{1} a n d \frac{x}{1} = \frac{y}{2} = \frac{z}{3}$ perpendicular to each other.

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Solution

The equations of the given lines are $\frac{x - 5}{7} = \frac{y + 2}{- 5} = \frac{z}{1} a n d \frac{x}{1} = \frac{y}{2} = \frac{z}{3}$ direction ratios of the given lines are respectively (7, - 5,1) and (1, 2, 3).
Two lines with direction ratios, $a_{1}, b_{1}, c_{1} a n d a_{2}, b_{2}, c_{2}$ are perpendicular to each other, if
$a_{1} a_{2} + b_{1} b_{2} + c_{1} c + 2 = 0$
$∴$ (7) (1) + (-5) (2) + (1) (3) = 7 - 10 + 3 = 0
Therefore, the given lines are perpendicular.

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Similar questions

Q. Show that the lines

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

and

\frac{x}{1} = \frac{y}{2} = \frac{z}{3}

are perpendicular to each other.

Q. Show that the lines

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

are perpendicular to each other.

Q. Show that the lines

\frac{x - 3}{2} = \frac{y + 1}{- 3} = \frac{z - 2}{4}

and

\frac{x + 2}{2} = \frac{y - 4}{4} = \frac{z + 5}{2}

are perpendicular to each other.

Q. Show that the lines:

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

and

\frac{x - 2}{1} = \frac{y - 4}{3} = \frac{z - 6}{5}

.
intersect each other. Find the co-ordinates of intersecting point also.

Q. The value of

k

so that the lines

\frac{x - 1}{- 3} = \frac{y - 2}{2 k} = \frac{z - 3}{2}

\frac{x - 1}{3 k} = \frac{y - 5}{1} = \frac{z - 6}{- 5}

are perpendicular to each other, is:

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Show that the lines x−57=y+2−5=z1 and x1=y2=z3 perpendicular to each other.

Show that the lines $\frac{x - 5}{7} = \frac{y + 2}{- 5} = \frac{z}{1} a n d \frac{x}{1} = \frac{y}{2} = \frac{z}{3}$ perpendicular to each other.