Unformatted text preview: 1.2. VECTORS AND THEIR ARITHMETIC 23 →
−
v
+→
−
w century; the ﬁrst elementary exposition was given by Edwin Bidwell
Wilson (18791964) in 1901 [54], based on lectures by the American
mathematical physicist Josiah Willard Gibbs (18391903)3 [17].
By a geometric vector in R3 (or R2 ) we will mean an “arrow” which can
be moved to any position, provided its direction and length are
maintained.4 We will denote vectors with a letter surmounted by an arrow,
→
like this: − . We deﬁne two operations on vectors. The sum of two vectors
v
→
is formed by moving − so that its “tail” coincides in position with the
w
→
− , then forming the vector − + − whose tail coincides with
→→
“head” of v
v
w
→
− and whose head coincides with that of − (Figure 1.18). If
→
that of v
w
→
−
w →
−
v
Figure 1.18: Sum of two vectors
→
instead we place − with its tail at the position previously occupied by the
w
→
− and then move − so that its tail coincides with the head of − ,
→
→
tail of v
v
w
→→
− + − , and it is clear that these two conﬁgurations form a
we form w
v
parallelogram with diagonal
→→→→
− +− =− +−
v
w
w
v
(Figure 5.18). This is the commutative property of vector addition.
A second operation is scaling or multiplication of a vector by a
number. We naturally deﬁne
→→
1− = −
v
v
→→→
2− = − + −
v
v
v
→→→→
− = − + − + − = 2− + −
→→
3v
v
v
v
v
v and so on, and then deﬁne rational multiples by
→
→
→
→
− = m − ⇔ n − = m− ;
w
v
w
v
n
3 I learned much of this from Sandro Caparrini [6, 7, 8]. This narrative diﬀers from the
standard one, given by Michael Crowe [10]
4
This mobility is sometimes expressed by saying it is a free vector. ...
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 Calculus, Addition, Vectors, Josiah Willard Gibbs, Michael Crowe, Edwin Bidwell, ﬁrst elementary exposition, Sandro Caparrini

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