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Scalar Multiple  Theorem 3
If A is a vector with components a_{1} and a_{2} and c is a scalar, then the scalar multiple cA is the vector with components ca_{l}, ca_{2}. Notice that the product of a scalar and a vector is a vector. Geometrically, for positive c, cA is the vector in the same direction as A whose length is c times the length of A (Figure 10.1.15). (  c)A is the vector in the opposite direction from A whose length is cA. We sometimes write Ac for cA, and A/c for (l/c)A. Figure 10.1.15 Scalar Multiples In physics, Newton's second law of motion states that F = mA where F is the force vector acting on an object, A is the acceleration vector, and the scalar m is the mass of the object. In economics, if all prices are increased by the same factor c due to inflation, then the new price vector Q will be a scalar multiple of the initial price vector P, Q = cP. THEOREM 3 Let A and B be vectors and s, t be scalars. (i) 0A = 0. 1A = A, (s)A = (sA). (ii) Scalar Associative Law s(tA) = (st)A. (iii) Distributive Laws (s + f)A = sA + fA, s(A + B) = sA + sB. (iv) sA = sA. We shall prove only part (iv) which says that the length of sA is s times the length of A. Let A have components a_{1}, a_{2}. Then sA has components sa_{1}, sa_{2}. Thus
A unit vector is a vector U of length one. The two most important unit vectors are the basis vectors i and j. i, the unit vector along the xaxis, has components (1, 0). j, the unit vector along the .vaxis, has components (0, 1). Figure 10.1.16 shows i and j.
Figure 10.1.16: Basis vectors A vector can be conveniently expressed in terms of the basis vectors.


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