Copyright © 2011 Pearson, Inc.
6.2Dot Product of Vectors
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What you’ll learn about
The Dot Product Angle Between Vectors Projecting One Vector onto Another Work
… and why
Vectors are used extensively in mathematics and science applications such as determining the net effect of several forces acting on an object and computing the work done by a force acting on an object.
Copyright © 2011 Pearson, Inc. Slide 6.2 - 3
Dot Product
The dot product or inner product of u u1,u
2
and v v1,v
2 is
uv u1v
1u
2v
2.
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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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Example Finding the Dot Product
Find the dot product.
4,3 1, 2
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Example Finding the Dot Product
4,3 1, 2 (4)( 1) (3)( 2) 10
Find the dot product.
4,3 1, 2
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Angle Between Two Vectors
If is the angle between the nonzero vectors u and v,
then
cos uvu v
and cos-1 uvu v
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Example Finding the Angle Between Vectors
Let u 2,3 and v 4, 1 .
Find the angle between the vectors u and v.
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Example Finding the Angle Between Vectors
cos uvu v
= 2,3 4, 1
2,3 4, 1
5
13 17
So,
cos 1 3
13 1
70¼
Let u 2,3 and v 4, 1 .
Find the angle between the vectors u and v.
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Orthogonal Vectors
The vectors u and v are orthogonal if and only if u·v = 0.
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Projection of u and v
If u and v are nonzero vectors, the projection of u
onto v is
projvu
uv
v2
v.
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Work
If F is a constant force whose direction is the same as
the direction of AB, then the work W done by F in
moving an object from A to B is W | F || AB |