Solid Geometry & Vectors

Recognize nonzero direction vectors and find them from two points, parametric equations, or two independent perpendicular directions.

Distinguish point membership, line containment, segments and lines, and learn the limits of perspective drawings.

Distinguish unique intersection, disjoint parallelism, and containment, and prove line containment from two distinct points.

Prove both directions of the perpendicular-plane criterion and apply it to a pyramid with a right-triangle base.

Use two intersecting lines and transfer perpendicularity through parallelism, preserving all independence conditions.

Apply parallel-plane ratios with finite intersections, preserve point order in the converse, and construct a fixed plane parallel to a moving line.

Use two intersecting in-plane directions to prove plane parallelism, and distinguish independent directions from parallel lines.

Check both direction and noncontainment when proving line–plane parallelism, with tetrahedron midsegment applications.

Combine a verified common point with a parallel direction, while checking distinct planes and coincident-line cases.

Distinguish a fixed line from a fixed point and prove concurrence for a family of plane intersections.

Place points in two distinct planes, check intersection existence, and allow intersections on extended triangle sides.

Construct the actual polygonal section, distinguish auxiliary intersections on extensions, and verify edge membership and cyclic order.

Reduce a spatial intersection to two coplanar lines and check uniqueness, parallelism, and containment.

Classify distinct and coincident lines using coplanarity and common points, with rigorous skew-line proofs.

Find and verify two distinct common points, distinguish planes from faces, and handle parallel base-line exceptions.

Apply the noncollinearity, off-line point, intersecting-line and distinct-parallel-line criteria for a unique plane.

Construct the common perpendicular with two planes and prove both incidence conditions, perpendicularities, and uniqueness.

Prove the correct perpendicular foot, calculate pyramid heights, and preserve zero-distance and nondegeneracy conditions.

Find a perpendicular foot, justify minimal distance, and distinguish distances to a line and a segment.

Determine when a perpendicular plane through a line is unique, construct its section, and verify interior vertices.