By Herbert S Green
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Extra info for Matrix mechanics
The definition may be stated as follows: the component vectors of A are the products of the components of A and the basis vectors. 4 Calculus of vectors. Two dimensions 37 two-dimensional Cartesian coordinate-system, we have Ax = Ax ex , Ay = Ay ey . 7, offer a clear picture of the link between the component vectors Ax , Ay and the (scalar) components Ax , Ay of a vector A. We will now define addition and subtraction of vectors. Consider two vectors A and B. The vector-sum of A and B is a new vector, which we shall designate by C.
Let a vector A be situated with its initial point at the origin, as shown in Fig. 7. 7: Vector components Then the components Ax and Ay are the coordinates of the terminal point of A. They are not vectors, but ordinary numbers. Such numbers are called scalars. 4 Calculus of vectors. Two dimensions 36 might have called them ‘scalar components’. Note that the scalar components of a vector might be negative (for example Ax < 0, if the vector points in the negative x direction). The vector quantities Ax and Ay (as shown in Fig.
We have a Cartesian coordinate system. The basis vectors along the x-axis and y-axis, are written ex and ey , respectively. The magnitude of a vector A is designated by |A| . The basis vectors ex and ey have per definition magnitude equal to 1, |ex | = |ey | = 1. 4 Calculus of vectors. 4 Calculus of vectors. Two dimensions 35 The quantities Ax and Ay in Fig. 6 will be called the components of the vector A. (Note that in the calculus of vectors it is usual to use the upper right suffix to select a component rather than to indicate the exponent of a power.