Question

Find the vector equation of the plane passing through the points A(1, 0, 1), B(1, –1, 1) and C(4, –3, 2).

Answer 1

Let the p.v. of points A(1, 0, 1), B(1, –1, 1) and C(4, –3, 2) be

`veca = bari + bark`, `barb = bari - barj + bark` and `barc = 4bari - 3barj + 2bark`

`barb-bara = -barj`, `barc-bara = 3bari - 3barj + bark`

`(barb-bara)xx(barc-bara) = |(bari,barj,bark),(0,-1,0),(3,-3,1)| = -bari + 3bark`

Equation of plane through A, B, C in vector form is

`(barr-bara).[(barb-bara)xx(barc-bara)] = 0`

`(barr-bara).(-bari + 3bark) = 0`

`barr.(-bari+3bark) = (bari+bark).(-bari + 3bark) = -1+3 = 2`

∴ `barr.(-bari + 3bark) = 2`