Analytic geometry in space

Up to Year 11 we worked in the Cartesian plane: two axes, two coordinates. In Year 12 we make the dimensional leap and introduce the coordinate system in space: three mutually perpendicular axes $x,y,z$ (which respect the right-hand rule: thumb $\to x$, index $\to y$, middle finger $\to z$), and each point identified by an ordered triple $(x,y,z)$. Much of the intuition and the formulae of the plane carry over, but important conceptual novelties appear: the plane becomes an object (no longer the ambient space), the relative positions are richer (two lines can be skew) and a new tool is needed — the cross product — to work with perpendicularities in three dimensions. The chapter runs through coordinates and distances, vectors with their scalar and cross products, the equation of the plane and pencils of planes, the line, relative positions, the distance formulae and finally the sphere.

Sezioni

• Coordinates, points, distances
• Vectors in space
• The plane
• The line in space
• Relative positions of two lines
• Distances
• Relative positions line-plane and angles
• Sphere

Approfondimenti

• Thales in space

Esercizi

• All the exercises of the chapter