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Equation of a Plane Passing through a Point and Parallel to the Two Given Vectors

Consider the ...

Question

Consider the following equation of planes ,
$P_{1} : 2 x - 4 y + 6 z = 5 P_{2} : - x + 2 y - 3 z = 8 P_{3} : 3 x + 6 y + 3 z = 24.$
Then which of the following is/are correct

P_{1}

and

P_{2}

are parallel.

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P_{1}

and

P_{3}

are parallel.

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P_{2}

and

P_{3}

are perpendicular.

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P_{1}, P_{2}

and

P_{3}

are parallel.

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Solution

The correct option is C $P_{2}$ and $P_{3}$ are perpendicular.
Given planes:
$P_{1} : 2 x - 4 y + 6 z = 5 P_{2} : - x + 2 y - 3 z = 8 P_{3} : 3 x + 6 y + 3 z = 24.$
$D . r^{'} s$ of normal to plane $P_{1} = (2, - 4, 6)$
$D . r^{'} s$ of normal to plane $P_{2} = (- 1, 2, - 3)$ and
$D . r^{'} s$ of normal to plane $P_{3} = (3, 6, 3)$
As, $D . r^{'} s$ of planes $P_{1}$ and $P_{2}$ are proportional but not proportional to $P_{3}$
So, planes $P_{1}$ and $P_{2}$ are parallel.
And, for plane $P_{2}, P_{3} : a_{2} a_{3} + b_{2} b_{3} + c_{2} c_{3} = 0$
So, plane $P_{2}$ and $P_{3}$ are perpendicular.

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