What is a Vector?
Vector: A mathematical object that possesses both magnitude (size) and direction, describing a translation in space. Vectors are distinct from scalars, which have magnitude only.
Vectors are written using either bold notation or an arrow on top:
Think of a vector as an instruction: "move this far, in this direction." A displacement of 5 km north-east is a vector; a temperature of 5°C is a scalar.
The same vector can be placed anywhere in space , what matters is its length and direction, not its starting point. This is the free vector concept. Two arrows with identical length and direction represent the same vector, regardless of where they are drawn.
Position Vectors and Displacement Vectors
Position Vector: The position vector of a point P is the vector extending from the fixed origin O to the point P. It specifies the location of P in the coordinate system.
Displacement Vector: A displacement vector describes the directed path from point A to point B , it encodes how to get from A to B, independent of where A and B sit in space.
If point A has position vector and point B has position vector , then:
and in the reverse direction:
Imagine you're standing at point A on a map. Your position vector tells you how far and in what direction you are from the map's origin. The displacement vector tells you exactly how to walk from A to reach B , it doesn't care where A is on the map, only the journey between them.
A position vector is technically also a direction vector, but it references a specific point in space. Don't confuse a general free vector (which floats anywhere) with a position vector (which is anchored at the origin).