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Dot Products
Objectives of this Section
• Find the Dot Product of Two Vectors
• Find the Angle Between Two Vectors
• Determine Whether Two Vectors and Parallel or Orthogonal
• Decompose a Vector into Two Orthogonal Vectors
• Compute Work
as
defined is thevectors,
twoare and If 2211
wvproductdot
jiwjiv
baba
2121 bbaa wv
find ,4 and 52 If wvjiwjiv
1542 wv 8 5 3
Properties of Dot Product
If u, v, and w are vectors, thenCommutative Property
uvvu
Distributive Property
wuvuwvu
0
2
v0
vvv
Angle between Vectors
If and are two nonzero vectors, the angle
, 0 < , between and is determined
by the formula
u v
u v
vu
vu
cos
Find the angle between = 2 and
= 4 + 3 .
u i j
v i j
v u
vucos
5383142 vu
u 2 1 52 2
v 4 3 16 9 25 52 2
5
1
55
5cos
v u
vu
4.63
v i j 4 3
u i j 2
Determine whether the vectors v = -3i + 2j and w = -9i + 6j are parallel.
The vectors are parallel since w = 3v.
w v
wvcos 27 12
13 117 39
15211
0
Two vectors v and w are orthogonal if and only if 0wv
Determine whether the vectors v = 4i - j and w = 2i + 8j are orthogonal.
08124 wv
The vectors v and w are orthogonal.
w = 2i + 8j
v = 4i - jx
y
If and are two nonzero vectors, the
vector projection of onto is
v w
v w
ww
wvvw 2
proj
The decomposition of into and
where is parallel to and is
perpendicular to , is
1 2
1 2
v v v
v w v
w
,
1221 proj vvvww
wvvv w
v1
v2
v
w
Find the vector projection of = 2 + onto
= + . Decompose into two vectors
and , where is parallel to
and is orthogonal to .1 2 1
2
v i j
w i j v
v v v w
v w
ww
wvvv w 21 proj
ji
222 11
1112
32
i j
v v v2 1 2i j32
i j 12
12
i j
v i j 2
w i j
v i j132
v i j2
12
12
The work W done by a constant force F in moving an object from A to B is defined as
ABW F
Find the work done by a force of 30 pounds acting in the direction 3i + j in moving an object 20 feet from (0, 0) to (20, 0).
i
10
350
j
10
150 (3, 1)
(20, 0)
ABW F
jiji 02010
1
10
350
W
W 300010
foot - pounds
pounds-foot 68.948W
