2. Representations of vectors





Given points A and B, we may think of the position vector v = as the vector that acts on the point A to get B. If the line segments AB and DC have the same direction and the same length, then ABCD is a parallelogram and the position vectors and are equal; we write this as = . The vector v is a free vector because although it has a definite direction and length it does not have any particular position in space. When using vectors to model physical phenomena it is not always appropriate to use free vectors. For example, in order to completely describe the effect of a force we need to give a vector representing the magnitude and direction of the force as well as a point of application of the force. Once we have chosen a suitable point as origin we can describe the point of application by the position vector of the point relative to the origin. It is usual to think of the vector representing the force as being confined to a line and for this reason it is often called a line vector. Two line vectors are equal if they have the same direction and length and lie along the same line. 

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