Recap
o Vector addition: If u
u1, upl and U
v1, v2, then
e.g., 1,3 2,4
3, 7
Or
Vector negation: If
v1, v2, then
v2
Vector negation
Let U
Iv1 vol be a vector in R2. We
define its negative, denoted -i, by setting
(-1)U
v1
V2
Visually, this simply amounts to a reversal of direction.
Remark: Note that
for any E R2, we have
This justifies our notation.
Vector subtraction: Given vectors u u1, u2 and
U
v1, v2 in R2, we define their difference, denoted u v, by
setting
(-i)
In other words,
1, 2
Visually:
or
Recap o Vector addition: If u u1, upl and U v1, v2, then e.g., 1,3 2,4 3, 7 Or Vector negation: If v1, v2, then v2 Vector negation Let U Iv1 vol be a vector in R2. We define its negative, denoted -i, by setting (-1)U v1 V2 Visually, this simply amounts to a reversal of direction. Remark: Note that for any E R2, we have This justifies our notation. Vector subtraction: Given vectors u u1, u2 and U v1, v2 in R2, we define their difference, denoted u v, by setting (-i) In other words, 1, 2 Visually: orExample 1.6: If
2, 11 and U
3, 2, compute
v
and draw it in the ane.
Solution: W- V C2- (-3), 4-2 Cs,-11
(NB
Remark: Note that for any points A and B in the ay-plane,
we have
This is equivalent to the earlier formula OB OA+A
AB
OA
Example 1.7: If the vector 1,3 is drawn in the zy-plane
with initial point A (-2,-1), what is its terminal point?
Solution:
OA AB
OB
OB
AB
OA.
So, terminal point
B (-3,2)
Example: If u [1,3), D-2,4l, what is 2u 3i?
Solution
20 3 20,31-30-2,41
C2,61 -6,127
C2 6),
121
Eg,-61
Vectors in three dimensions
The preceding discussion readily extends into three dimen-
sions.
Definition: We write R3 or
the set of all ordered triples of
real numbers.
Example 1.6: If 2, 11 and U 3, 2, compute v and draw it in the ane. Solution: W- V C2- (-3), 4-2 Cs,-11 (NB Remark: Note that for any points A and B in the ay-plane, we have This is equivalent to the earlier formula OB OA+A AB OA Example 1.7: If the vector 1,3 is drawn in the zy-plane with initial point A (-2,-1), what is its terminal point? Solution: OA AB OB OB AB OA. So, terminal point B (-3,2) Example: If u [1,3), D-2,4l, what is 2u 3i? Solution 20 3 20,31-30-2,41 C2,61 -6,127 C2 6), 121 Eg,-61 Vectors in three dimensions The preceding discussion readily extends into three dimen- sions. Definition: We write R3 or the set of all ordered triples of real numbers.The elements of R3 are again called vectors, and we express
them using either row or column notation:
a1
a1, a2, a3
Or
a2
a3
There is again a visual perspective: We introduce a "right-
handed coordinate system" with three mutually perpendicular
axes labelled ar, y and 2. The vector la
a2, a3 should be
nterpreted as representing the displacement from the origin
O to the point (a1, a2, a3) in this coordinate system:
a, x displacement
displacement
az
at
ay
z displacement
More generally, the displacement from a point A a1, a2, a3)
to another point B (b1, b2, b3) determines a vector
a2, b3
a3
Arithmetic: The arithmetical concepts of the previous sec-
tion extend in the obvious way: Let il
01, 2, ugl,
v1, v2, v3 be vectors in R3 and let c e R be a scalar.
Do 0,01 (zero vector)
ii+ if vi, u2 v2, v13 val (addition).
ci cvi, cv2, curs] scalar multiplication
-if vi, v2, -val (negation).
ii-U lu1 -v1, u2 v2, us v3) subtraction.
The visual interpretation is the same as before, and previous
identities (e
AB BC AC, AB OB -OA) remain
valid.
Vectors in n dimensions
We now generalize to the case of n dimensions, where n is
any natural number. Since the case n 24 escapes our visual
sense, we

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