#### 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.

#### Solution

QA and QB are the perpendiculars drawn from the point Q(a,b,c)to YZ and ZX planes

`therefore A=(0, b,c) and B=(a,0,c)`

The required plane is pasing through O(0, 0, 0), A(0, b, c) and B(a, 0, c)

The vector equation of the plane passing thorugh the O, A, B is

`bar r.(bar(OA)xxbar(OB))=bar 0.(bar(OA)xxbar(OB))`

let `bar r.(bar xxbar b)=0`

Now `bar(OA)=bara=0.hati+bhatj+chatk`

`bar(OB)=barb=ahati+0hatj+chatk`

`therefore bar(OA)xxbar(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`