A cross-section perpendicular to a line in a pyramid
Construct a plane through A perpendicular to SC, prove all four section vertices lie on edges, and check the coordinate parameter.
Solid Geometry & Vectors
Explore explanations, worked examples and practice in this topic. Review the prerequisites in each lesson before starting.
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The three perpendiculars theorem: hypotheses and both directions
Relate perpendicularity to a projected line, prove both implications with vector decomposition, and isolate point-image cases.
Angle between two planes: definition and coordinate formula
Use nonzero normal vectors and an absolute-value dot product to find the prescribed angle from 0 to 90 degrees.
Cross-sections parallel to two given directions
Construct a trapezoidal section face by face, prove all four vertices lie on edges, and handle degenerate endpoint positions.
Distance between skew lines in 3D
Derive the distance formula using an auxiliary plane and scalar triple product, and handle intersecting or parallel exceptions.
Parallel lines in 3D: distinguishing them from coincident lines
Test parallel directions and point membership to distinguish distinct parallel lines, coincident lines, and skew lines.
The angle between a line and a plane
Define the angle by orthogonal projection, derive the sine formula, and check cube and coordinate examples with all special cases.
Calculating angles between lines in space using vectors
Use nonzero direction vectors and the absolute-value dot-product formula, with checked cube diagonal examples.
Perpendicular directions of lines in space
Test perpendicular directions with dot products, prove the regular-tetrahedron result, and distinguish orthogonal directions from intersection.
The dot product in space: lengths, angles, and perpendicularity
Calculate dot products, sum and difference lengths, and perpendicular directions while treating zero vectors separately.
Vector angles and line angles in space
Distinguish vector angles from line angles, preserve zero-vector exclusions, and apply given angles in a rhombus-base parallelepiped.
Coplanar vectors: definition and linear-combination criterion
Translate vectors to a common origin, test coplanarity, and distinguish an independent plane basis from dependent or zero-vector cases.
Representing vectors: three-point, parallelogram, and parallelepiped rules
Express diagonal vectors through edges, preserve signs and vertex order, and distinguish identities from unique basis representations.
Vector addition, subtraction, and negative vectors in space
Use head-to-tail addition and subtract componentwise, preserving endpoint order and distinguishing vectors from their lengths.
Vector length and equality in space
Calculate vector lengths and distinguish equality, parallelism, equal lengths, and negative vectors.
Vectors in space: endpoints and direction
Read directed segments, distinguish parallel vectors from the same direction, and handle zero vectors and translations.
Cross-sections parallel to a plane
Construct a section parallel to a lateral face, check its edge ratios, and distinguish interior and endpoint cases.