The definition of the product of two vectors is
Transcript of The definition of the product of two vectors is
2121 bbaa wv
1542 wv
8 5 3
The definition of the product of two vectors is:
2211 , and , where baba wv
This is called the
dot product.
Notice the answer
is just a number
NOT a vector.
find ,1,4 and 5,2 If wvwv
Properties of the Dot Product• Let u, v, and w be vectors in the plane or in space
and let c be a scalar.
2
1. u v v u
2. 0 v 0
3. u (v w) u v u w
4. v v v
5. c(u v) cu v u cv
If and are two nonzero vectors, the angle
, 0 < , between and is determined
by the formula
u v
u v
vu
vu
cos
The dot product is useful for several things. One of
the important uses is in a formula for finding the angle
between two vectors that have the same initial point.
u
v
.4,3= and 12, =between angle theFind vu
v u
vu cos 5383142 vu
u 2 1 52 2
v 4 3 16 9 25 52 2
5
1
55
5cos
v u
vu
3,4v
1,2u
4.635
1cos 1
Find the angle between the vectors
v = <3, 2> and w = <6, 4>
The vectors have the same
direction. We say they are
parallel because remember
vectors can be moved around
as long as you don't change
magnitude or direction.
w v
wv cos
5213
818
676
26 1
01cos 1 What does it mean when the
angle between the vectors is 0?
2,3v
4,6w
Orthogonal (Perpendicular) Vectors• Two vectors are orthogonal if their dot product is 0
• Example: u v 0
Let u 2, 3 and v 3,2
u v (2)(3) ( 3)(2) 0
so these vectors are othogonal
0 wv
Determine whether the vectors v = 4i - j and
w = 2i + 8j are orthogonal.
08124 wvThe vectors v and
w are orthogonal.
If the angle between 2 vectors is , what would their dot
product be? 2
v u
vu cos
Since cos is 0, the
dot product must be 0.
2
2
2
Vectors u and v in this case are called orthogonal.
(similar to perpendicular but refers to vectors).
compute their dot product
and see if it is 0
w = 2i + 8j
v = 4i - j
The work W done by a constant force F in
moving an object from A to B is defined
as
ABW F
A use of the dot product is found in the formula below:
This means the force is in
some direction given by the
vector F but the line of
motion of the object is along
a vector from A to B