Question

If from a point Q (a, b, c) perpendiculars QA and QB are drawn to the YZ and ZX planes respectively, then find the vector equation of the plane QAB.

Answer 1

QA and QB are the perpendiculars from the point Q( a, b, c) to YZ and ZX planes.
∴ A = ( 0, b, c) and B = (a, 0, c)
The required plane is passing through O(0, 0, 0), A(0, b, c) and B(a, 0, c)
The vector equation of the plane passing through the O,A,B is
`barr.( bar[OA] xx bar[OB] ) = bar0.( bar[OA] xx bar[OB] )` 
i.e; `barr.(veca xx vecb ) = 0`
Now, `bar[OA] = bara = 0.hati + bhatj + chatk`

and `bar[OB] = barb = ahati + 0.hatj + chatk`

∴ `bar(OA) xx bar(OB) = |[hati, hatj, hatk], [0, b, c], [a, 0, c]|`

= `(bc - 0)hati - (0 - ac)hatj + (0 - ab)hatk`

= `bchati + achatj - abhatk`

∴ from (1), the vector equation of the required plane is
`barr.(bchati + achatj - abhatk ) = 0`