Geometry in space
Geometry in space for IB Analysis and Approaches HL: vectors in R3, scalar and vector products, lines, planes, angles and distances.
Geometry in space takes to three dimensions what you already know from the plane: vectors have three components, lines are described by a point and a direction, and planes appear. With the scalar product you measure angles and check perpendicularity, and with the vector product you obtain a vector perpendicular to two others, as well as areas and volumes. Then you calculate angles, distances and relative positions (how lines and planes meet, relying on the systems from topic 7). It is one of the topics with the most weight at Higher Level: it appears in Paper 1, in Paper 2 and often in long context problems (aeroplanes, drones, buildings).
Recommended order
Follow the sections in order: vectors, scalar product, vector product, lines and planes, angles and distances, relative positions and volumes and areas of solids. At the end you will find IB exam questions grouped by type. Making a sketch before calculating avoids most of the mistakes.
Plane geometry (topic 5) and systems of linear equations (topic 7).
Sections in this topic
Vectors in space
Vectors in R3: components, magnitude, addition, scalar multiplication, parallel vectors, midpoint and base vectors i, j, k, with IB AA HL examples.
3 worked examples Section 2Scalar product and the angle between vectors in space
Scalar product in R3: cosine and component formulas, properties, angle between vectors, perpendicularity, projections and parameters. IB AA HL.
3 worked examples Section 3Vector product, areas and volumes
Vector (cross) product in R3: calculation with components and determinant, properties, perpendicular vectors, areas and volumes (IB AA HL).
3 worked examples Section 4Equations of lines and planes in space
Equations of lines and planes in R3: vector, parametric and symmetric forms, normal vectors and Cartesian equations, with IB AA HL examples.
3 worked examples Section 5Angles and distances in space
Angles between lines and planes, and distances from a point to a line or plane and between skew lines, with worked examples for IB AA HL.
3 worked examples Section 6Relative positions and intersections of lines and planes
Relative positions of two lines, a line and a plane, and two or three planes in space, with intersection points and lines. Worked examples for IB AA HL.
3 worked examples Section 7Volumes and surface areas of solids
Volume and surface area of prisms, cylinders, pyramids, cones and spheres, composite solids, diagonals and angles inside a solid. Worked examples for IB AA HL.
3 worked examples End of topicIB exam questions: topic 8
IB exam questions on geometry in space: vectors, lines, planes, angles, distances and relative positions, with step-by-step solutions.
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