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 ca_l, 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.

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.