Question

The direction cosines of any normal to the xy-plane are

• A. 1, 0, 0
• B. 0, 1, 0
• C. 1, 1, 0
• D. 0, 0, 1

Hint:

Direction cosines of coordinate axes: x-axis \((1, 0, 0)\), y-axis \((0, 1, 0)\), z-axis \((0, 0, 1)\).

Answer 1

The Correct Option is D

The problem is to find the direction cosines of any normal to the xy-plane. To solve this, let's delve into the concept of direction cosines and the geometry of the coordinate planes: 

Concepts:

• The xy-plane is defined by the equation \(z = 0\). Any normal to this plane is along the z-axis.
• Direction cosines of a vector are the cosines of the angles made by the vector with the coordinate axes (x, y, and z).

Step-by-step Explanation:

1. A normal to the xy-plane can be represented by the vector along the z-axis, which has components (0, 0, 1).
2. The angle made by this vector with the:
   • x-axis is 90 degrees (cos(90) = 0),
   • y-axis is 90 degrees (cos(90) = 0),
   • z-axis is 0 degrees (cos(0) = 1).
3. Thus, the direction cosines of this normal vector are (0, 0, 1).

Conclusion:

The direction cosines of any normal to the xy-plane are (0, 0, 1), which matches the correct answer.

Therefore, the correct answer is:

0, 0, 1