Reshebnik Ryabushko. Solved IDZ 3.1, Option 1

1. Given four points A1 (x1, y1, z1), A2 (x2, y2, z2), A3 (x3, y3, z3), A4 (x4, y4, z4)
Make equations:
a) plane A1A2A3; b) straight A1A2;
c) straight line A4M, perpendicular to the plane A1A2A3;
d) straight line A3N parallel to straight line A1A2;
e) a plane passing through point A4 perpendicular to straight line A1A2.
Calculate:
f) sine of the angle between straight line A1A4 and plane A1A2A3;
g) cosine of the angle between the Oxy coordinate plane and the A1A2A3 plane;
1.1 A1 (3, 1, 4), A2 (−1, 6, 1), A3 (−1, 1, 6), A4 (0, 4, −1)

2. Solve the following tasks
2.1 Find the values ​​of the segments cut off on the coordinate axes by the plane passing through the point M (−2, 7, 3) parallel to the plane x - 4y + 5z - 1 = 0

3. Solve the following tasks
3.1 Prove the parallelism of lines (x-1) / 6 = (y + 2) / 2 = z / -1 and x - 2y + 2z - 8 = 0, x + 6z - 6 = 0