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

Answer 1

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