# Dot Product of Vectors

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```					   Section 6-2

Dot Product of Vectors
Section 6-2
the dot product
finding the angle between two
vectors
orthogonal vectors
projection of a vector onto another
vector
work problems
The Dot Product
the dot product of a vector has many
physical and geometric applications
you cannot multiply two vectors
together, but the dot product can be

u = u1 , u2   and v = v1 , v2
u v = u1 v1 + u2 v2
Properties of Dot Product

2
u v=v u                 u u= u
u (v + w) = u v + u w
( cu ) v = u ( cv ) = c ( u v )
Find   −3 , 5   1, 4
Angle Between Two Vectors
the angle, , between two vectors, u
and v, can be found using the
following formula

cos θ =
u v
or
u v
 u v 
θ = cos 
−1

 u v 
Find the angle between the vectors:
u = 2 ,1    and v = −1 , 3
Orthogonal Vectors
if   = 90 then the dot product of
the two vectors will be zero (because
cos 90 = 0)
this leads to the following definition
of “orthogonal vectors”

vectors u and v are orthogonal if u v = 0
Orthogonal Vectors
orthogonal means nearly the same
thing as perpendicular
the difference is because of the zero
vector – a vector cannot be
perpendicular to the zero vector
(because no angle is formed)
however, any vector is orthogonal to
the zero vector because the dot
product will be zero
Prove the vectors u = 2 , 3 and v = −6 , 4 are orthogonal
A sled is sitting on the side of a hill inclined at 45 .
The weight of the sleigh and its contents is 140 lbs.
What force is required to keep the sled from sliding
down the hill?
Work
if an object is moved from point A to
point B by a force, F, then we can
compute the amount of Work that is
done
the farther the object must be
moved, the more work must be done
the unit for work is typically foot-
pounds
Work

if therforce, F, is in the same direction
as AB then the amount of work, W,
uuu
to move the object from A to B is:
r
uuu
W = F AB
Work
if the force, F, is in any other
direction, then the work, W, to move
the object from A to B is:

W = F AB cos θ
r
uuu

where θ is the angle between F and AB
r
uuu
Find the work done by a 15 lb force acting in the direction
2 , 3 in moving an object 3 feet from (0 , 0) to (3 , 0)

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