4
Vectors Vectors To add two vectors (geometrically) put them head to tail. The resultant vector carries the tail of the first to the head of the second
Vecto
rsV
ectors
To
add
two
vecto
rs (geo
metrically
) pu
t them
head
to tail.
Th
e resultan
t vecto
r carries the tail o
f the first to
the h
ead o
f
the seco
nd
v
u
2v
.3v
u/2
or .5
uor
u2 1
1.2
uor
u4 5
Vecto
r Op
eration
s
It mak
es sense to
add
vecto
rs. To
add
two
vecto
rs
(geo
metrically
) pu
t them
head
to tail.
Th
e resultan
t vecto
r carries the tail o
f the first to
the h
ead o
f
the seco
nd
A v
ector is a d
irection
with
a mag
nitu
de, so
yo
u can
thin
k o
f it
as an arro
w th
at can h
ave an
y startin
g p
oin
t.
It mak
es sense to
mu
ltiply
a real nu
mb
er by
a vecto
r. Th
is just
chan
ges th
e leng
th o
f (or “scales”) th
e vecto
r with
ou
t
chan
gin
g its d
irection
. Fo
r this reaso
n, w
e call real nu
mb
ers
“scalars.”
Vecto
r Op
eration
s
On
ce we h
ave v
ector ad
ditio
n an
d scalar m
ultip
lication
, we
can d
efine v
ector su
btractio
n b
y u
–v
= u
+ (–
v).
vu
–v
u–v
v
u–v
Tw
o V
iews o
f Vecto
r Su
btractio
n
2. T
his is th
e same as p
uttin
g u
and
vtail to
tail and
draw
ing
the v
ector fro
m th
e head
of v
to th
e head
of u
1. u
–v
is u+
(–v
), so p
ut th
e tail of –
vo
n th
e head
of u
and
draw
the v
ector fro
m th
e tail of u
to th
e head
of –
v.
Stan
dard
Un
it Vecto
rs
v ui
j
Ad
din
g V
ectors
v
ui
j
4i
2j
3i
–j
uv
4i
3i
2j –
j
Scalin
g V
ectors
4i
2j
2u
4j
8i
u
Su
btractin
g V
ectors
v
ui
j
4i
2j
3i
–j
u
–v
4i
–3i
2j j
Stan
dard
Un
it Vecto
rs in 3
-D
w
ij
k
Vecto
r Op
eration
s
To
add
two
vecto
rs, add
the co
rrespo
nd
ing
amo
un
ts of i
and
j
(and
k.)
Each
vecto
r is a bit o
f ian
d a b
it of j
(in 2
-dim
ensio
ns) o
r a
bit o
f i, j, and
k (in
3-d
imen
sion
s)
To
scale a vecto
r, mu
ltiply
each co
mp
on
ent b
y th
e scalar.
Tw
o su
btract tw
o v
ectors, su
btract th
e com
po
nen
ts.
Len
gth
s of V
ectors
Un
it Vecto
rs or “D
irection
s”U
nit V
ectors o
r “Directio
ns”
Fin
d v
ector w
ith m
agn
itud
e 5 in
the d
irection
of ⟨1
, –1
/2, 1
/2⟩
