Question

Determine the equation of the line passing through the point (–1, 3, –2) perpendicular to the lines:

`x/1 = y/2 = z/3` and `(x + 2)/(-3) = (y - 1)/2 = (z + 1)/5`

Answer 1

Let the equation of required line be,

`(x - x_1)/a = (y - y_1)/b = (z - z_1)/c`

∵ The line passes through (–1, 3, –2).

∴ `(x + 1)/a = (y - 3)/b = (z + 2)/c`   ...(i)

Now, given lines are

`x/1 = y/2 = z/3`

And `(x + 2)/(-3) = (y - 1)/2 = (z + 1)/5`

Since, the required line is perpendicular to these two lines

∴ a × 1 + b × 2 + 3 × c = 0

⇒ a + 2b + 3c = 0   ...(ii)

And a × (–3) + b × 2 + c × 5 = 0

⇒ –3a + 2b + 5c = 0   ...(iii)

Solving equations (ii) and (iii) using cross-multiplication method,

`a/(10 - 6) = b/(-9 - 5) = c/(2 + 6)`

⇒ `a/4 = b/(-14) = c/8 = k`   ...(say)

⇒ a = 4k, b = –14k and c = 8k.

Putting these values in equation (i), we get

`(x + 1)/(4k) = (y - 3)/(-14k) = (z + 2)/(8k)`

⇒ `(x + 1)/2 = (y - 3)/(-7) = (z + 2)/4`,

which is the required equation of line.