 

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 = , 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 u_{1} + u_{2} + u_{3} + ...... + 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.
