Vectors are fundamental mathematical objects that possess both magnitude and direction, describing a translation. Hence they showcase both quantity and orientation in some space.
They can be written as a bold letter or with a vector sign on top, for example
$$\vec{v} \text{ OR } \mathbf{v}$$
Position vectors represent the location of a point in space relative to a fixed origin. For instance, if we consider a point P in three-dimensional space, its position vector $\vec{OP}$ would extend from the origin O to the point P.
Displacement vectors, on the other hand, describe the change in position from one point to another. If we have two points A and B, the displacement vector $\vec{AB}$ represents the directed path from A to B. The displacement vector from B to A would be $\vec{BA}$.
Let the right red position vector be $\vec{OA}=\mathbf{a}$ and the left red position vector be $\vec{OB} = \mathbf{b}$. Then the blue vector goes from $\vec{OA}$ to $\vec{OB}$ which is $\vec{AB}$ and can be calculated as $\mathbf{b}- \mathbf{a}$
A position vector can be viewed as a direction vector that specifies a particular point. That is why we write $\mathbf{b}-\mathbf{a}$, where $\mathbf{a}$ and $\mathbf{b}$ are position vectors, instead of using position notation.
Vectors are often visually represented as arrows, where the length of the arrow corresponds to the vector's magnitude, and the direction of the arrow indicates the vector's direction.
The same vector can be represented by any arrow with the same length and direction, regardless of its starting point. This is known as the "free vector" concept.
In three-dimensional space, we define three standard unit vectors:
These vectors form a basis for 3D space, meaning any vector can be expressed as a linear combination of these base vectors.
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