# Properties of Vectors

- Vectors can exist at any point in space.
- Vectors have both direction and magnitude.
- Any two vectors that have the same direction and magnitude are equal, no matter where in space they are located; this is called
*vector equality*. **Unit Vector**A vector has both magnitude and direction. The magnitude of is a scalar written as A or . A unit vector along is defined as a vector whose magnitude is unity and its direction is along .**Component Vectors**Any vector in Cartesian coordinates may be represented as or,*A*,_{x}*A*,_{y}*A*are called the component vectors in_{z}*x*,*y*and*z*directions respectively.**Vector Decomposition**Choosing a coordinate system with an origin and axes, we can decompose any vector into component vectors along each coordinate axis. In figure ,we choose Cartesian coordinates. A vector at*P*can be decomposed into the vector sum,*x*-component vector pointing in the positive or negative*x*-direction, is the*y*-component vector pointing in the positive or negative*y*-direction, and is the*z*-component vector pointing in the positive or negative*z*-direction (Figure).

**Vector Decomposition**

**Direction Angles and Direction Cosines of a Vector**The direction cosines of a vector are merely the cosines of the angles that the vector makes with the*x*,*y*, and*z*axes, respectively. We label these angles*Î±*(angle with the*x*-axis),*Î²*(angle with the*y*-axis), and*Î³*(angle with the*z*-axis).