Unformatted text preview: From the above, the negative of a and the
product of the scalar —1 and the vector a are
both the vector which has the same magni—
tude as a. but the direction is the opposite of
that of a. Hence we have —a = (—1)a.
Figure 6: Scalar Multiplication. We have already seen the commutative and associative laws of vector additions. There are some
more important properties of scalar multiplication and vector additions. Let a and b be vectors, A
and p. be real numbers. Mua) = (Apa)a, (Associative law of multiplication by a scalar)
(A + Ma 2 Aa + ,ua, (Scalar distributive law)
Ma + b) = Aa + Ab. (Vector distributive law) The vector distributive law for the case that A > 0 follows from properties of similar triangles and Figure 7. The proof for the other cases and the proofs of the other two laws are left as exercises
for the 1141 students. ...
View Full Document