The floor of the banquet hall in a hotel is made of polished stone. There is a large chandelier attached to the ceiling of the hall. Consider floor of the hotel as a plane having the equation x - y + z = 4 and chandelier is suspended at the point (1, 0, 1) from the wall.
On the basis of above information, answer the following questions.
The length of the perpendicular from the point (1, 0, 1) to the plane x - y + z = 4 is
Answer & explanation
Correct answer: option 2
The correct answer is option 2: $\frac{2}{\sqrt{3}}$
We need the distance of the point (1, 0, 1) from the plane
x − y + z = 4
1. Write plane in standard form
x − y + z − 4 = 0
Here
a = 1, b = −1, c = 1, d = −4
Point = (1, 0, 1)
2. Distance formula
Distance = |ax₁ + by₁ + cz₁ + d| / √(a² + b² + c²)
3. Substitute values
Numerator: |1(1) + (−1)(0) + 1(1) − 4|
= |1 + 0 + 1 − 4|
= |−2|
= 2
Denominator:
√(1² + (−1)² + 1²)
= √(1 + 1 + 1)
= √3
4. Distance = 2 / √3