Question

Find the Cartesian equations of the line passing through the point A(1, 1, 2) and perpendicular to the vectors `barb = hati + 2hatj + hatk and barc = 3hati + 2hatj - hatk`.

Answer 1

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

It is perpendicular to the vectors `barb = hati + 2hatj + hatk and barc = 3hati + 2hatj - hatk`

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

∴ p + 2q + r = 0, 3p + 2q – r = 0

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

∴ `p/(-4) = q/(4) = r/(-4)`

∴ `p/(-1) = q/(1) = r/(-1)`

∴ The required line has direction ratios –1, 1, –1

The cartesian equations of the line passing through (x₁, y₁, z₁) 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)/(-1)`,

i.e. x  – 1 =  y – 1 = z  – 2