## Scalar Multiple - Theorem 3

If A is a vector with components a1 and a2 and c is a scalar, then the scalar multiple cA is the vector with components cal, ca2. 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 c|A|. 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| = |s||A|.

We shall prove only part (iv) which says that the length of sA is |s| times the length of A.

Let A have components a1, a2. Then sA has components sa1, sa2.

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 x-axis, has components (1, 0). j, the unit vector along the .v-axis, 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.

Last Update: 2006-11-07