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**Unformatted text preview: **25 1.2. VECTORS AND THEIR ARITHMETIC →
−
u →
−
w →
−
v
→→→
− =− +−
u
v
w →
−
u →
-−
w →
−
v
→→→
− =− −−
v
u
w Figure 1.20: Diﬀerence of vectors →
→→
→→
→
associative: − + (− + − ) = (− + − ) + − .
u
v
w
u
v
w
• Multiplication of vectors by scalars →→
→
→
distributes over vector sums: r (− + − ) = r − + r − , and
v
w
w
v
→
− = r − + s− .
→
→
distributes over scalar sums: (r + s) v
v
v We will explore some of these properties further in Exercise 3.
The interpretation of displacements as vectors gives us an alternative way
→
to represent vectors. We will say that an arrow representing the vector −
v
is in standard position if its tail is at the origin. Note that in this case
the vector is completely determined by the position of its head, giving us a
→
natural correspondence between vectors − in R3 (or R2 ) and points
v
−
→
→
3 (resp. R2 ). − corresponds to P if the arrow − P from the origin to
P ∈R
v
O
→
→
P is a representation of − : that is, − is the vector representing that
v
v
→
3 which moves the origin to P ; we refer to − as the
displacement of R
v
position vector of P . We shall make extensive use of the correspondence
between vectors and points, often denoting a point by its position vector
→
− ∈ R3 .
p
Furthermore, using rectangular coordinates we can formulate a numerical
speciﬁcation of vectors in which addition and multiplication by scalars is
−
→ −→
very easy to calculate: if − = OP and P has rectangular coordinates
v
→
(x, y, z ), we identify the vector − with the triple of numbers (x, y, z ) and
v
→
− = (x, y, z ). We refer to x, y and z as the components or entries
write v
−
→
→ −→
of − . Then if − = OQ where Q = (△x, △y, △z ) (that is,
v
w
→
− = (△x, △y, △z )), we see from Figure 1.21 that
w
→→
− + − = (x + △x, y + △y, z + △z );
v
w that is, we add vectors componentwise. ...

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