The Dot Product Angles Between Vectors Orthogonal Vectors
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Transcript of The Dot Product Angles Between Vectors Orthogonal Vectors
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The Dot ProductThe Dot ProductAngles Between VectorsAngles Between VectorsOrthogonal VectorsOrthogonal Vectors
The beginning of Section 6.2a
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Definition: Dot Product
The dot product or inner product of u = u , uand v = v , v is
1 21 2
u v = u v + u v1 21 2
The sum of two vectors is a… vector!vector!
The product of a scalar and a vector is a… vector!vector!
The dot product of two vectors is a… scalar!scalar!
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Properties of the Dot Product
Let u, v, and w be vectors and let c be a scalar.
1. u v = v u
2. u u = |u| 2
3. 0 u = 0
4. u (v + w) = u v + u w
(u + v) w = u w + v w
5. (cu) v = u (cv) = c(u v)
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Finding the Angle BetweenTwo Vectors
vu
v – u
0
2 2v v v u u v u u u v 2 u v cosθ
2 2v u v u u v 2 u v cosθ
2u v 2 u v cosθ
2 2 2 2v 2u v u u v 2 u v cosθ
2 2 2v u u v 2 u v cosθ
u vcosθ
u v
1 u v
θ cosu v
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Theorem:Angle Between Two Vectors
vu
v – u
01 u v
θ cosu v
If 0 is the angle between nonzerovectors u and v, then
u vcosθ
u v
and
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Definition: Orthogonal Vectors
The vectors u and v are orthogonal ifand only if u v = 0.
The terms “orthogonal” and “perpendicular”The terms “orthogonal” and “perpendicular”are nearly synonymous (with the exceptionare nearly synonymous (with the exception
of the zero vector)of the zero vector)
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Guided Practice
Find each dot product
1. 3, 4 5, 2
2. 1, –2 –4, 3
3. (2i – j) (3i – 5j)
= 23= 23
= –10= –10
= 11= 11
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Guided Practice
Use the dot product to find the length ofvector v = 4, –3 ((hint: use property 2!!!)hint: use property 2!!!)
Length = 5Length = 5
2v v v
v v v 4 4 3 3
25
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Guided Practice
Find the angle between vectors u and v
0 = 55.4910 = 55.491
u = 2, 3 , v = –2, 5
u v 2 2 3 5 112 2u 2 3 13 2 2v 2 5 29
1 11θ cos
13 29
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Guided Practice
Find the angle between vectors u and v
u cos i sin j3 3
5 5v 3cos i 3sin j
6 6
1 3,2 2
3 3 3,
2 2
01 3 3 3 3
u v2 2 2 2
3 3 3 3
4 4
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Guided Practice
Find the angle between vectors u and v
u cos i sin j3 3
5 5v 3cos i 3sin j
6 6
1 3,2 2
3 3 3,
2 2
0 = 900 = 90
Is there an easier way to solve this???Is there an easier way to solve this???
1θ cos 0
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Guided Practice
Prove that the vectors u = 2, 3 andv = –6, 4 are orthogonal
u v = 0!!!u v = 0!!!
u v 2 6 3 4 12 12 0 Check the dot product:
Graphical Support???Graphical Support???
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First, let’s look at a brain exercise…Page 520, #30:
Find the interior angles of the triangle with vertices (–4,1),(1,–6), and (5,–1).
Start with a graph…
A(–4,1)
B(5,–1)
C(1,–6)
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First, let’s look at a brain exercise…
A(–4,1)
B(5,–1)
C(1,–6)
4 1,1 6CA 11111111111111
5 1, 1 6CB 11111111111111
5,7
4,5
5 4 7 5CA CB 1111111111111111111111111111
15
2 25 7CA 11111111111111
74 2 24 5CB 11111111111111
41
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First, let’s look at a brain exercise…
A(–4,1)
B(5,–1)
C(1,–6)
1 15cos
74 41C
74.197
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First, let’s look at a brain exercise…
A(–4,1)
B(5,–1)
C(1,–6)
9,2BA 11111111111111
4, 5BC 11111111111111
26BA BC 1111111111111111111111111111
85BA 11111111111111
41BC 11111111111111
1 26cos
85 41B
63.869
180 74.197 63.869A 41.934
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Definition: Vector Projection
The vector projection of u = PQ onto a nonzero vector v = PSis the vector PR determined by dropping a perpendicular fromQ to the line PS.
u
P
Q
SR
Thus, u can be broken into components PR and RQ:
u = PR + RQ
v
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Definition: Vector Projection
u
P
Q
SR
Notation for PR, the vector projection of u onto v:
PR = proj uv
The formula:
proj u = vvu v|v| 2
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Some Practice Problems
Find the vector projection of u = 6, 2 onto v = 5, –5 .Then write u as the sum of two orthogonal vectors, oneof which is proj u.v Start with a graph… v 2u proj u u
u v 6 5 2 5 20
22v 5 5 50
v 2
u vproj u v
v
205, 5
50 2, 2
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Some Practice Problems…
Find the vector projection of u = 6, 2 onto v = 5, –5 .Then write u as the sum of two orthogonal vectors, oneof which is proj u.v
u = proj u + u = 2, –2 + 4, 4v 2
Start with a graph…v 2u proj u u
2 vu u proj u
6,2 2, 2
4,4
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Some Practice Problems…
Find the vector projection of u = 3, –7 onto v = –2, –6 .Then write u as the sum of two orthogonal vectors, oneof which is proj u.v
u = proj u + u = –1.8,–5.4 + 4.8,–1.6v 2
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Some Practice Problems…
Find the vector v that satisfies the given conditions:2u = –2,5 , u v = –11, |v| = 10
1 2v ,v v
1 22 5 11v v 2 21 2 10v v
A system to solve!!!
1 2
5 11
2 2v v
22
2 2
5 1110
2 2v v
2 22 2 2
25 110 12110
4 4 4v v v
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Some Practice Problems…
Find the vector v that satisfies the given conditions:2u = –2,5 , u v = –11, |v| = 10
1 2v ,v v
22 2
29 110 810
4 4 4v v
2 22 2 2
25 110 12110
4 4 4v v v
22 229 110 81 0v v
2 21 29 81 0v v 2
811,29
v
v 3, 1 OR 43 81
,29 29
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Some Practice Problems…
Now, let’s look at p.520: 34-38 even:
What’s the plan???What’s the plan??? If u v = 0 If u v = 0 orthogonal! orthogonal!If u = If u = kkv v parallel! parallel!
34) Neither
36) Orthogonal
38) Parallel