6.2 Notes.notebook
6.2 The Dot Product
January 27, 2016
Name: _________________
Objectives: Students will be able to calculate dot products
and projections of vectors.
Dot Product: The dot product or inner product of
u = <u1,u2> and v = <v1,v2> is u v = u1v1+u2v2.
Note: The dot product is not a vector but a real number!
Examples Find each dot product.
1.) <3,4> <5,2>
2.) <1,-2> <-4,3>
3.) (2i-j) (3i-5j)
Jan 11­9:27 AM
Property Let u = <u1,u2>. Find u u.
Examples Use the dot product to find u . Let u = <4,-3>.
Jan 11­9:55 AM
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u v
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Angle Between Two Vectors
If θ is the angle between two vectors u and v, then cosθ= u v .
u v
Examples Find the angle θ between the vectors.
1.) u = <2,3> and v = <-2,5>
2.) u = cos(π/3)i + 2sin(π/3)j
v = 3cos(5π/6)i + 3sin(5π/6)j
Jan 11­9:30 AM
Note: Orthogonal = Perpendicular
Orthogonal Vectors The vectors u and v are orthogonal
if and only if u v = 0.
Example Prove that u = <2,3> and v = <-6,4> are orthogonal.
Jan 11­10:03 AM
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Projecting One Vector Onto Another
If u and v are nonzero vectors, the projection of u onto v is:
projvu = u v v
v 2
Jan 11­10:55 AM
Example Find the vector projection of u = <6,2> onto v = <5,-5>.
Then write u as the sum of two orthogonal vectors, one of which
is projvu.
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Example Find the interior angles of the triangle with the
vertices (-4,1), (1,-6) and (5,-1).
Jan 11­11:24 AM
Examples Determine whether the vectors are parallel,
perpendicular or neither.
1.) u = <2,5> , v = <10/3, 4/3>
2.) u = <5,-6> , v = <-12,-10>
Homework: pages 519-520 #1-37 odd
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