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Title: Dot Product
Description: This will help you understand Dot Product in vector calculus (Math)
Description: This will help you understand Dot Product in vector calculus (Math)
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CHAPTER 1
1
...
Given ⃗ =< 2, −4 > and ⃗ =< −1, 6 >
u
v
Solution:
⃗ • ⃗ = 2(−1) + (−4)(6) = −26
u v
2
...
θ
x
u
1
y
§ Example
What is the angle between ⃗ =< 1, 2, 3 > and w =< −2, 3, 4 >?
v
⃗
Solution
⃗ • w = (1)(−2) + (2)(3) + (3)(4) = 16
v ⃗
And also
⃗•w =
v ⃗
⃗ w cos θ
v ⃗
√
=
(1)2 + (2)2 + (3)2 ·
= (20
...
15) cos θ = 16
16
cos θ =
= 0
...
15
θ = 37
...
y
α
Ax
x
ˆ
i
The direction cosine of a non zero vector are :
cos α =
Ax
,
⃗
A
cos β =
Ay
,
⃗
A
cos γ =
Az
,
⃗
A
It can be shown that
cos2 α + cos2 β + cos2 γ
2
2
2
Ax
Ay
Az
=
+
+
⃗
⃗
⃗
A
A
A
=
=
A2 + A2 + A2
x
y
z
⃗
A
⃗
A
2
⃗
A
2
2
=1
§ Example
ˆ
Find the direction angles of ⃗ = 3ˆ + ˆ − 2k
...
802,
α = 0
...
267,
(3)2 + (1)2 + (−2)2
cos γ = √
−2
(3)2 + (1)2 + (−2)2
β = 1
...
535,
γ = 2
...
The diagram
below shows the projection of a vector (blue) onto a line
...
Question:
How to construct a vector which has the same direction with vector ⃗ and its magnitude
u
is equal to the projection of vector ⃗ onto ⃗ ?
v
u
⃗
u
⃗
v
Reference for direction
...
We have
v
to specify the direction of the line somehow, so we’ll assume there’s a vector ⃗ which
u
gives the direction of the line
...
e
...
u
u v
⃗
u
⃗
v
Reference
...
Since ⃗ − proj⃗ (⃗ ) is orthogonal to ⃗ ,
u
v
u
u v
u v
we have that
Vector component
of v orthogonal to u
⃗ − proj⃗ (⃗ )
v
u v
⃗
v
...
Find
v
i j
u i
j
a) Vector projection of ⃗ along ⃗
v
u
b) Vector projection of ⃗ orthogonal to ⃗
v
u
c) Scalar projection of ⃗ onto ⃗
v
u
Solution:
5
⃗
u
|⃗ |
u
)
=
⃗ •⃗
v u
⃗ •⃗
v u
⃗ •⃗ ⃗
v u u
·(1) =
·
=
|⃗ | |⃗ |
u
u
⃗
u
⃗
u
a) proj⃗ (⃗ ) =
u v
2
3
< 1, 2, 2 >
b) ⃗ − proj⃗ (⃗ ) =
v
u v
1
3
< 4, −7, 5 >
c) scalar projection of ⃗ onto ⃗ , proj⃗ ⃗ = 2
v
u
uv
Summary
• A vector can be written as
ˆ
⃗
V =< v1 , v2 , v3 >= v1ˆ + v2ˆ + v3 k
i
j
• Direction angle
cos α =
v1
⃗
v
,
cos β =
v2
⃗
v
,
cos γ =
v3
⃗
v
,
satisfying the relation cos2 α + cos2 β + cos2 γ = 1
Title: Dot Product
Description: This will help you understand Dot Product in vector calculus (Math)
Description: This will help you understand Dot Product in vector calculus (Math)