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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 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`
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