# Properties of Vectors

1. Vectors can exist at any point in space.
2. Vectors have both direction and magnitude.
3. 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.
4. 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 .
5. Component Vectors Any vector  in Cartesian coordinates may be represented as  or,

where Ax, Ay, Az are called the component vectors in xy and z directions respectively.
6. 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,

where  is the 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
1. 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 xy, 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).