Position of a vector in a three dimensional plane

Position of a Vector in a Two-Dimensional Plane or Space

In vector algebra, every point in space can be represented by a vector originating from a fixed reference point called the origin. This directed line segment is referred to as the position vector.

Position Vector of a Point

Consider O as the origin of the coordinate plane, and A as a point located in two-dimensional space. The position vector of point A is denoted as:
\( \overrightarrow{a} = \overrightarrow{OA}\)

This vector points from the origin O to the point A and represents both the direction and location of point A in space.

Position Vectors of Two Points

Assume two points, A and B, each associated with their respective position vectors:
\( \overrightarrow{a} = \overrightarrow{OA}\)

The vector from point A to point B is denoted by &overrightarrow;AB and is given by:
\(\overrightarrow{OB} = \overrightarrow{b} - \overrightarrow{a}\)

In other words, moving from point A to point B can be described by taking the difference between the position vectors of B and A.

Vector Equation Relationship

From the above, we can also express this relationship:
\(\overrightarrow{OA}+ \overrightarrow{AB} = \overrightarrow{OB}\)

This vector equation confirms that if we move from O to A (via \(\overrightarrow{OA}\) and then from A to B (via \(\overrightarrow{AB}\), the total movement takes us from O to B (via \(\overrightarrow{AB}\).

Properties of Vector Addition

Vector addition follows specific properties that are essential in geometry and physics:

• Commutative Property: \(\overrightarrow{a} + \overrightarrow{b} = \overrightarrow{b} + \overrightarrow{a}\)
• Associative Property: \((\overrightarrow{a} + \overrightarrow{b}) + \overrightarrow{c} = \overrightarrow{a} + (\overrightarrow{b} + \overrightarrow{c})\)
• Identity Element (Zero Vector): \(\overrightarrow{a} + \overrightarrow{0} = \overrightarrow{a}\), meaning adding the zero vector leaves any vector unchanged.
• Existence of Additive Inverse: \(\overrightarrow{a} + (-\overrightarrow{a}) = \overrightarrow{0}\)

The position vector provides a powerful tool to represent points in space. By understanding relationships such as:
\(\overrightarrow{AB} = \overrightarrow{b} - \overrightarrow{a}\) and \(\overrightarrow{OB} + \overrightarrow{AB} = \overrightarrow{OB}\),
we can apply the fundamental rules of vector addition and subtraction to describe motion, geometry, and forces in two- or three-dimensional space.