School of Mathematics and Statistics, The University of Sydney
 8. Vector algebra
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Glossary
Examples

Vector algebra in geometric form

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We discuss properties of the two operations, addition of vectors and multiplication of a vector by a scalar. We do this first for free vectors.

Equality of Vectors
Two vectors u and v are equal if they have the same magnitude (length) and direction.
The Negative of a Vector
The negative of the vector u is written -u, and has the same magnitude but opposite direction to u. If u = - --> AB, then -u = ---> BA.
Commutative Law of Addition
u + v =  v + u

for all vectors v and u.

Associative Laws
(u + v) + w  = u + (v + w)     s(tu) = t(su) = (st)u
for all vectors u, v and w and for all scalars s and t.

As an illustration of the first of these associative laws, we translate the three vectors u, v and w so that they are drawn head to tail, and then draw (u + v) + w in the first figure below and u + (v + w) underneath it, demonstrating that both equal ---> OC.

                                   C                                     w               (u + v) + w                                 B                u + v                             v  O             u          A

                                   C                                     w               u + (v + w)                         v + w   B                               v  O             u          A

We may then simply write u + v + w, without using brackets. This associative law extends to sums of any number of vectors taken in any order, so that the expression u1 + u2 + u3 + ...... + un is well defined.

Distributive Laws
s(u + v) = su +  sv (s + t)u = su +  tu
for all vectors v and u and for all scalars s and t.

The first of these distributive laws is illustrated below in the case s = 2.

       2(u + v)   u + v          v    u

 2u + 2v             2v    2u

Laws Involving the Zero Vector
u +  0 = 0 + u = u      u -  u = 0,
for all vectors u.

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