6.4 Vectors and Dot Products
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Transcript of 6.4 Vectors and Dot Products
6.4 Vectors and Dot Products
Finding the angle between two vectors
Writing a vector as the sum of two vectors components
Definition of Dot Product
Given: Two vectors in Component form
The result is a number, not a vector
2211
2121 ,,
vuvuvu
vvvanduuu
Find the Dot Product
Given 3,26,4
Find the Dot Product
Given
26188
3624
3,26,4
Products of the Dot Product
cvuorvcuvuc
scalariscwhere
vvv
wuvuwvu
v
uvvu
2
00
The Angle Between two vectors
For angles 0
vu
vuCos
The Angle Between two vectors
For angles
Find the angle between
0
vu
vuCos
2,3,5,1 vu
2222 23,)5(1
132531
2,3,5,1
vu
vu
vuvu
vuCos
457071.0
...7071.0
338
13
1326
13
23,)5(1
132531
2,3,5,1
1
2222
Cos
Cos
Cos
vu
vu
vu
vu
vuCos
The Angle Between two vectors
For angles
Vectors are Orthogonal if there Dot Product (u●v)= 0What is the angle between the vectors,
Why?
0
vu
vuCos
Definition of Vector Components
Let u and v be nonzero vectors.
u = w1 + w2 and w1 · w2 = 0
Also, w1 is a scalar of v
The vector w1 is the projection of u onto v, So w1 = proj v u
w 2 = u – w 1
vv
vuuprojv
2
Decomposing of a Vector Using Vector Components
vv
vuuprojw
wwu
vu
v
21
21
2,6,5,3
Decomposing of a Vector Using Vector Components
5
2,5
62,6
40
8
2,626
2)5(63
2,6,5,3
1
222
1
21
21
w
w
vv
vuuprojw
wwu
vu
v
Decomposing of a Vector Using Vector Components
5
2,5
62,6
40
8
2,626
2)5(63
2,6,5,3
1
222
1
21
21
w
w
vv
vuuprojw
wwu
vu
v
5
27,5
9
5
2,5
6
5
27,5
9
5
25,
5
63
2
2
u
w
w
Definition of Work
Work is force times distance.
If Force is a constant and not at an angle
If Force is at an angle
PQFW
PQFprojWPQ
Homework
Page 447 – 448
# 1, 5, 17, 21, 25,
29, 33, 37, 41,
45, 49, 53, 63, 67
Homework
Page 447 – 448
# 4, 8, 12, 16,
20, 24, 28, 32,
36, 40, 52