Question

Find Cartesian equation of the line passing through the point A(2, 1, −3) and perpendicular to vectors `hat"i" + hat"j" + hat"k"` and `hat"i" + 2hat"j" - hat"k"`

Answer 1

Let `bar"b" = hat"i" + hat"j" + hat"k"` and `bar"c" = hat"i" + 2hat"j" - hat"k"`

We know that `bar"b" xx bar"c"` is perpendicular to both `bar"b"` and `bar"c"`.

∴ `bar"b" xx bar"c" = |(hat"i", hat"j", hat"k"),(1, 1, 1),(1, 2, -1)|`

= `hat"i"(-1, -2) -hat"j"(-1, -1) + hat"k"(2 - 1)`

= `-3hat"i" + 2hat"j" + hat"k"`

∴ The direction ratios of the required line are – 3, 2, 1 and it passes through A(2, 1, – 3).

∴ The Cartesian equation of a line passing through the point (x₁, y₁, z₁) and having direction ratios (a, b, c) is

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

i.e., `(x - 2)/(-3) = (y - 1)/2 = (z + 3)/1`