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HW… WebAssign 12.3 (19 problems, 108 min.)
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MAT 1236Calculus III
Section 12.3 The Dot Product
http://myhome.spu.edu/lauw
HW… WebAssign 12.3 (19 problems, 108 min.)
Preview Define a new operation on vectors. Angle(s) between vectors Projection of vectors
• Formula• Why the formula makes sense? (More
important in a long run)
We are Interested in … The angle between two forces The component of one force along the
direction of another force
Vectors The angle between two forces
Dot ProductIf and , the dot product of a and b is the number
1 2 3 1 2 3
1 1 2 2 3 3
, , , ,a b a a a b b b
a b a b a b
1 2 3, ,a a a a 1 2 3, ,b b b b
Example 1(a)
(b)
1,0, 1 0,1, 1
2 3i j k i k
Dot product and Length
1 2 3 1 2 3, , , ,a a a a a a a a
Properties
Properties
Geometric Meaning0
a
b a b
a
b a b
2v v v
Formula0
a
b
cos a ba b
PPFNE Derive the formula of the angle between
two vectors
Example 2Find the angle between the given vectors
1, 1,1 , 1, 1,2a b
cos a ba b
Special Case: Orthogonal VectorsTwo vectors are orthogonal if the angle between them is a right angle.
cos 0a ba b
Example 3Find the value of x such that the given vectors are orthogonal.
, , 1 , 1, ,6a x x b x
ProjectionVector Projection
Scalar Projection
2aa bproj b aa
a
b
aproj b
acomp b
aa bcomp ba
Remark
Scalar Projection
aa bcomp ba
a
b
aproj b
acomp b
2
Make Sense?Vector Projection
How do I know the vector is in the same direction of a?
2aa bproj b aa
a
b
aproj b
acomp b
02
Make Sense?Vector Projection
How do I know <2,2> is in the same direction of <1,1>?
2aa bproj b aa
a
b
aproj b
acomp b
02
Make Sense?Vector Projection
Does the “length” of the vector agree with what we know?
2aa bproj b aa
aa bcomp ba
a
b
aproj b
acomp b
02
Proof: We need to recall…Vector Projection
2aa bproj b aa
a
b
aproj b
acomp b
Unit Vectors A unit vector is a vector with length is 1 e.g.
Unit VectorsThe unit vector in the same direction as vector
u
b
Unit VectorsUnit vector in the same direction as vector
So,
A vector equals to its length times the unit vector along the same direction
u
b
Proof:
a
b
x
Example 4Find the vector projection of onto .
2,3 , 4,1a b
2aa bproj b aa
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