Solution IDZ Ryabushko 3.1 - Option 12

Section: Chapter 3. Planes and Lines
Topic: Plane and line in space
Description: Drawing up equations of a plane and a straight line according to various conditions: three points, a point and a normal, a point and a direction vector, perpendicularity, parallelism. Calculation of angles between a straight line and a plane, between planes.

Tasks:
1. Given four points A₁(x₁, y₁, z₁), A₂(x₂, y₂, z₂), A₃(x₃, y₃, z₃), A₄(x₄, y₄, z₄). Make up the equations: a) plane A₁A₂A₃; b) straight line A₁A₂; c) straight line A₄M perpendicular to plane A₁A₂A₃; d) straight line A₃N parallel to straight line A₁A₂; e) plane passing through point A₄ perpendicular to straight line A₁A₂. Calculate: f) the sine of the angle between straight line A₁A₄ and plane A₁A₂A₃; g) the cosine of the angle between the coordinate plane Oxy and plane A₁A₂A₃.
2. Find the values of the segments cut off on the coordinate axes by a plane passing through point M parallel to a given plane.
3. Prove the parallelism of lines.

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