Representing and Naming Vectors
Simple Explanation
A vector is represented geometrically by an arrow: the arrow's length shows the vector's magnitude, and the arrow's direction shows the vector's direction. It can be named by its two endpoint letters with an arrow on top — →AB, read "vector AB," running FROM A TO B — or by a single bold lowercase letter, e.g. a.
Why Do We Need It?
Reading and writing vector notation correctly is essential before combining vectors in any calculation — mixing up →AB and →BA (which point in exactly opposite directions) is one of the most common early errors.
See It
An arrow starting at point A and ending at point B, labelled vector a
Worked Example
Reverse the direction of a named vector
Vector →PQ is 8 units long, pointing due east. Describe vector →QP.
Why Does This Work?
Reversing a vector's start and end letters moves the arrowhead to the other endpoint, which by definition reverses the direction while leaving the length (magnitude) unchanged. This is exactly what "the negative of a vector" means, and it's the idea behind vector subtraction.
Real-Life Example
Walking directions between two landmarks
Instructions say "walk from the library to the park" versus "walk from the park to the library."
Both journeys cover the same path length, but →LP (library to park) and →PL (park to library) point in exactly opposite directions — just like reversing vector notation.
Practice
Vector a has magnitude 6 and points due north. What is true of vector −a?
EasyCommon mistake
Writing →AB when →BA is meant (or vice versa) — always read vector notation as "from the first letter to the second letter."
Quick Review
- A vector is drawn as an arrow: length = magnitude, direction = direction.
- →AB runs from A to B; →BA is its reverse (same length, opposite direction).
- Vectors can also be named with a single bold lowercase letter, e.g. a.