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 direction cosine of the normal to the plane x - y + z = 4 are:

$\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}$

The correct answer is Option 1: $\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}$

Step 1: Identify the normal vector

For the plane
x − y + z = 4

The normal vector is given directly by the coefficients of x, y, z:

Normal vector = (1, −1, 1)

Step 2: Find its magnitude

Magnitude = √(1² + (−1)² + 1²)
= √(1 + 1 + 1)
= √3

Step 3: Direction cosines

Direction cosines of the normal are the components divided by the magnitude:

l = 1 / √3
m = −1 / √3
n = 1 / √3

Final Answer

Direction cosines = (1/√3, −1/√3, 1/√3)