Find the Cartesian equations of the line passing through the point A(1, 1, 2) and perpendicular to the vectors `bar"b" = hat"i" + 2hat"j" + hat"k" and bar"c" = 3hat"i" + 2hat"j" - hat"k"`

#### Solution

Let the required line have direction ratios p, q, r.

It is perpendicular to the vector `bar"b" = hat"i" + 2hat"j" + hat"k" and bar"c" = 3hat"i" + 2hat"j" - hat"k"`.

∴ it is perpendicular to lines whose direction ratios are 1, 2, 1 and 3, 2, – 1.

∴ p + 2q + r = 0, 3 + 2q – r = 0

∴ `p/|(2, 1),(2, -1)| = q/|(1, 1),(-1, 3)| = r/|(1, 2),(3, 2)|`

∴ `p/(-4) = q/(4) = r/(-1)`

∴ `p/(-1) = q/(1) = r/(-1)`

∴ the required line has direction ratios –1, 1, –1.

The cartesian equations of the line passing through (x_{1}, y_{1}, z_{1}) and having direction ratios a, b, c are `(x = x_1)/a = (y - y_1)/b = (z - z_1)/c`

∴ the cartesian equation of the line passing through the point (1, 1, 2) and having directions ratios –1, 1, – 1 are `(x - 1)/(-1) = (y - 1)/(1) = (z - 2)/(-2)`.