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Question
If the foot of the perpendicular drawn from the origin to the plane is (4, −2, -5), then the equation of the plane is ______
Options
4x + y + 5z = 14
4x − 2y − 5z = 45
x − 2y − 5z = 10
4x + y + 6z = 11
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Solution
4x − 2y − 5z = 45
Explanation:
O ≡ (0, 0, 0)
P ≡ (4, -2, -5)
`vec"a"=4hat"i"-2hat"j"-5hat"k"`
`vec"n"=vec"op"=4hat"i"-2hat"j"-5hat"k"`
⇒ `vec"PR".vec"n"=0`
⇒ `[vec"r"-vec"a"].vec"n"=0`
⇒ `vec"r".vec"n"-vec"a".vec"n"=0`
⇒ `vec"r".(4hat"i"-2hat"j"-5hat"k")-(4hat"i"-2hat"j"-5hat"k").(4hat"i"-2hat"j"-5hat"k")=0`
⇒ `vec"r".(4hat"i"-2hat"j"-5hat"k")-[16+4+25]=0`
⇒ `vec"r"=xhat"i"+"y"hat"j"+"z"hat"k"`
R ≡ (x, y, z)
⇒ `(xhat"i"+"y"hat"j"+"z"hat"k").(4hat"i"-2hat"j"-5hat"k")=45`
⇒ 4x - 2y - 5z = 45
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