§5 Vectors
Vectors are mathematical objects that capture both magnitude and direction. By representing geometric quantities — displacements, forces, velocities — as vectors, we gain powerful algebraic tools for solving geometry problems that would be difficult to handle with coordinates or classical methods alone.
Learning Objectives
By the end of this chapter you will be able to:
- Recognize vectors as directed line segments and determine when two vectors are equal, opposite, or collinear
- Express a vector by its coordinates given its endpoints, and compute its magnitude
- Add, subtract, and multiply vectors by scalars both geometrically and via coordinates
- Apply the collinearity theorem and the segment-division formula to solve geometric problems
- Compute the scalar (dot) product of two vectors and use it to find angles and check perpendicularity