Chapter 9: Vectors in Two and Three Dimensions
Vectors represent quantities with both magnitude and direction — force, velocity, displacement. This chapter covers vector algebra in 2D and 3D, dot products, cross products, and 3D geometry. Vectors are the language of physics, engineering, and computer graphics.
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Sections
9.1Vectors in Two Dimensions
A vector is a directed line segment. We represent vectors as ⟨a, b⟩, add them component-wise, and scale them by multiplying components. The magnitude uses the distance formula.
9.2The Dot Product
The dot product u · v is a scalar computed from components. It encodes the angle between two vectors and is used to find projections, determine orthogonality, and compute work in physics.
9.3Three-Dimensional Coordinate Geometry
Extend the Cartesian plane to 3D by adding a z-axis. Points become (x, y, z) triples. The distance formula uses all three differences, and spheres replace circles as the fundamental round shape.
9.4Vectors in Three Dimensions
3D vectors extend all 2D vector concepts with an added k component. The cross product — new to 3D — gives a vector perpendicular to both inputs, with magnitude equal to the parallelogram area and direction given by the right-hand rule.
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