The University of Sydney - School of Mathematics and Statistics

Vector algebra in geometric form	Page 1 of 2

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 = , then -u = .

Commutative Law of Addition

for all vectors v and u.

Associative Laws

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 .

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 + ...... + u_n is well defined.

Distributive Laws

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.

Laws Involving the Zero Vector

for all vectors u.