Co Ordinate Geometry

Centroid of triangle :
Given: , vertices , , .
x-coordinate:

Recompute using vector method: Centroid formula holds. Correct x-coordinate:

possible error in problem. Use given centroid.

y-coordinate:
z-coordinate:
Answer: , .

For pair of lines , the angle between them is:
For : , , .
For , let the angle be the same:
Square both sides:
Answer: Shown.

The equation of a plane with normal direction ratios and passing through is:
Given: Point , normal direction ratios .
Answer: .

The equation represents a pair of lines. Let one line be , i.e., .
Substitute into :
Since ,
Answer: .

Rewrite: .
The equation represents two lines:
1. .
2. .
Answer: , .

Direction ratios: Difference between points (2, 1, -3) and (0, 0, 0):
Line equation through origin (0, 0, 0) with direction ratios (2, 1, -3): Answer: .