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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 Description: 23268.png has both magnitude and direction. The magnitude of Description: 23275.png is a scalar written as A or Description: 23285.png. A unit vector Description: 23292.png along Description: 23299.png is defined as a vector whose magnitude is unity and its direction is along Description: 23307.png.
  5. Component Vectors Any vector Description: 23386.png in Cartesian coordinates may be represented as Description: 23395.png or,
    Description: 23405.png
    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,
    Description: 23448.png
    where Description: 23457.png is the x-component vector pointing in the positive or negative x-direction, Description: 23464.png is the y-component vector pointing in the positive or negative y-direction, and Description: 23472.png 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 α (angle with the x-axis), β (angle with the y-axis), and γ (angle with the z-axis).

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