Let the equation of the line which can be drawn from the point $(2, - 1, 3)$ , which is perpendicular to the lines $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}$ and $\frac{x - 4}{4} = \frac{y}{5} = \frac{z + 3}{3}$ be $\frac{x - 2}{k} = \frac{y + 1}{- m} = \frac{z - 3}{n}$ . Find $k - m + n$ .

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Solution

d.r s of line which is perpendicular to the lines $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}$ and $\frac{x - 4}{4} = \frac{y}{5} = \frac{z + 3}{3}$ is $(2 i + 3 j + 4 k) \times (4 i + 5 j + 3 k) = - 11 i + 10 j - 2 j$
Since, the line passes through $(2, - 1, 3)$
equation of lines is $\frac{x - 2}{11} = \frac{y + 1}{- 10} = \frac{z - 3}{2}$
$k = 11, m = 10, n = 2$
$k - m + n = 11 - 10 + 2 = 3$

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