College Trigonometry, 2011a

1016 Applications of Trigonometry
The properties in Theorem 11.18 are easily verified using the definition of vector addition. 9
the commutative property, we note that if ⃗v = 〈v 1 ,v 2 〉 and ⃗w = 〈w 1 ,w 2 〉 then
For
⃗v + ⃗w = 〈v 1 ,v 2 〉 + 〈w 1 ,w 2 〉
= 〈v 1 + w 1 ,v 2 + w 2 〉
= 〈w 1 + v 1 ,w 2 + v 2 〉
= ⃗w + ⃗v
Geometrically, we can ‘see’ the commutative property by realizing that the sums ⃗v + ⃗w and ⃗w + ⃗v
are the same directed diagonal determined by the parallelogram below.
⃗v
⃗w
⃗w + ⃗v
⃗v + ⃗w
⃗w
⃗v
Demonstrating the commutative property of vector addition.
The proofs of the associative and identity properties proceed similarly, and the reader is encouraged
to verify them and provide accompanying diagrams. The existence and uniqueness of the additive
inverse is yet another property inherited from the real numbers. Given a vector ⃗v = 〈v 1 ,v 2 〉, suppose
we wish to find a vector ⃗w = 〈w 1 ,w 2 〉 so that ⃗v + ⃗w = ⃗0. By the definition of vector addition, we
have 〈v 1 + w 1 ,v 2 + w 2 〉 = 〈0, 0〉, and hence, v 1 + w 1 = 0 and v 2 + w 2 =0. Wegetw 1 = −v 1 and
w 2 = −v 2 so that ⃗w = 〈−v 1 , −v 2 〉. Hence, ⃗v has an additive inverse, and moreover, it is unique
and can be obtained by the formula −⃗v = 〈−v 1 , −v 2 〉. Geometrically, the vectors ⃗v = 〈v 1 ,v 2 〉 and
−⃗v = 〈−v 1 , −v 2 〉 have the same length, but opposite directions. As a result, when adding the
vectors geometrically, the sum ⃗v +(−⃗v) results in starting at the initial point of ⃗v and ending back
at the initial point of ⃗v, or in other words, the net result of moving ⃗v then −⃗v is not moving at all.
⃗v
−⃗v
Using the additive inverse of a vector, we can define the difference of two vectors, ⃗v − ⃗w = ⃗v +(− ⃗w).
If ⃗v = 〈v 1 ,v 2 〉 and ⃗w = 〈w 1 ,w 2 〉 then
9 The interested reader is encouraged to compare Theorem 11.18 and the ensuing discussion with Theorem 8.3 in
Section 8.3 and the discussion there.