The Dot Product of Two Vectors The dot product of u and v in the plane is The dot product of u and v in space is Two vectors u and v are orthogonal ID: 570717
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Slide1
11.3
The Dot Product of Two VectorsSlide2
The
dot product
of
u and v in the plane is
The dot product of u and v in space is
Two vectors u and v are orthogonal if they meet at a right angle. if and only if u ∙ v = 0 (since slopes are opposite reciprocal)
(Read “u dot v”)
DefinitionsSlide3
ExamplesSlide4
Another form of the
Dot Product
:
PropertiesSlide5
Find the angle between vectors
u
and v:
ExamplesSlide6
Angles between a vector
v
and 3 unit vectors i, j
and k are called direction angles of
v, denoted by α, β, and γ respectively. Since
we obtain the following 3 direction cosines of
v: So any vector v has the normalized form:
Direction CosinesSlide7
Let
u
and
v be nonzero vectors.
w1 is called the vector component of u along v (or projection of u onto v), and is denoted by
projvu w2 is called the vector component of
u orthogonal to v
w
2
w
1
u
v
Vector ComponentsSlide8
A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60
o
north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they?
N
E
ApplicationSlide9
A Boeing 727 airplane, flying due
east at 500mph
in still air, encounters a 70-mph tail wind acting in the direction of 60o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they?
N
E
uSlide10
A Boeing 727 airplane, flying due east at 500mph in still air, encounters a
70-mph
tail wind acting in the direction of 60o
north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they?
N
E
v
u
60
oSlide11
A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60
o
north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they?
N
E
v
u
We need to find the magnitude and direction of the resultant vector u +
v
.
u+vSlide12
N
E
v
u
The component forms of
u
and v are:
u+v
500
70
Therefore:
and:Slide13
N
E
The new ground speed of the airplane is about 538.4 mph, and its new direction is about 6.5
o
north of east.
538.4
6.5oSlide14
1) Compute
2) Compute
4) Find the angle between vectors
v
and
w.3) List pairs of orthogonal and/or parallel vectors.
6) Find the projection of w onto u.5) Find the unit vector in the direction u.
7) Find vector component of w orthogonal to u.
Examples