The direction ratios of the line perpendicular to the linesx-72-y-17-3-z-61 and- x-51-y-32-z-4-2 are proportional toa 4-5-7b 4-5-7c 4-5-7d -4-5-7

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Equation of a Plane Parallel to a Given Plane

The direction...

Question

The direction ratios of the line perpendicular to the lines

$\frac{x - 7}{2} = \frac{y + 17}{- 3} = \frac{z - 6}{1} and, \frac{x + 5}{1} = \frac{y + 3}{2} = \frac{z - 4}{- 2}$ are proportional to
(a) 4, 5, 7
(b) 4, −5, 7
(c) 4, −5, −7
(d) −4, 5, 7

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Solution

(a) 4, 5, 7

We have

$\frac{x - 7}{2} = \frac{y + 17}{- 3} = \frac{z - 6}{1} \frac{x + 5}{1} = \frac{y + 3}{2} = \frac{z - 4}{- 2}$

The direction ratios of the given lines are proportional to 2, $-$ 3, 1 and 1, 2, $-$ 2.

The vectors parallel to the given vectors are $\vec{b_{1}} = 2 \hat{i} - 3 \hat{j} + \hat{k} and \vec{b_{2}} = \hat{i} + 2 \hat{j} - 2 \hat{k}$ .

Vector perpendicular to the given two lines is
$\vec{b} = \vec{b_{1}} \times {\vec{b}}_{2} = |\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & - 3 & 1 \\ 1 & 2 & - 2 \end{matrix}| = 4 \hat{i} + 5 \hat{j} + 7 \hat{k}$

Hence, the direction ratios of the line perpendicular to the given two lines are proportional to 4, 5, 7.

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The direction ratios of the line perpendicular to the lines x-72=y+17-3=z-61 and, x+51=y+32=z-4-2 are proportional to (a) 4, 5, 7 (b) 4, −5, 7 (c) 4, −5, −7 (d) −4, 5, 7

The direction ratios of the line perpendicular to the lines

$\frac{x - 7}{2} = \frac{y + 17}{- 3} = \frac{z - 6}{1} and, \frac{x + 5}{1} = \frac{y + 3}{2} = \frac{z - 4}{- 2}$ are proportional to
(a) 4, 5, 7
(b) 4, −5, 7
(c) 4, −5, −7
(d) −4, 5, 7