6.4 NOTES 6.4 NOTES
VECTOR OPERATIONS:
1. VECTOR ADDITION
2. SCALAR MULTIPLICATION
3. DOT PRODUCT Orthogonal Vectors
6.4 NOTES which work is correctly done? A
which work is incorrectly done? B
Find the magnitude of vector <2,­3>
A.
B.
6.4 NOTES PRE CALC 6.4 3 VECTOR OPERATIONS:
1. VECTOR ADDITION
2. SCALAR MULTIPLICATION 3. __________________
Dot Product
DOT PRODUCT Note: Dot Product is a number answer/scalar)
1. u v=
2. 0 v=
3. u (v+w)=
4.v v= v
5.c(u v)=
∨
∨
.
∨
∨
The dot product of u = u1, u2 and v = v 1, v2 is
u v = 2
Find the dot product :
.
.
1. 4,5 2,3
2. 2,­1 1,2
3. u = 3i + 2j
v = ­2i ­ 3j
23
0
­12
Notice that all of these answers are scalars!
Let u = ­1,3 , v = 2, ­4 , and w = 1,­2 Find each product:
.
4. (u v)w
.
5. u 2v
=
= (
= <­14,28>
­7
­28
6.4 NOTES Orthogonal Vectors
Vectors u and v are orthogonal if
(Orthogonal and perpendicular mean same thing i.e. meeting at right angles)
.
NOTE: The zero vector is orthogonal to every vector u, since 0 u = 0
Examples: Are the vectors orthogonal?
11. u = 2, ­3 and v = 6,4
4
2
­2
2
6
4
30
24
12. u = ­12, 30 and v = 5, 1
42
18
12
6
1
­12
­6
1
2
6.4 NOTES .
If the vectors are orthogonal, then u v = 0.
If vectors are parallel, then u and v are scalar multiples of each other.
Determine whether u and v are orthogonal, parallel, or neither:
1. u = 2,2
2. u = ­12i + 10j
v = ­8,­8
v = ­1 1
+
5i 6 j
= 0
3. u= <­3,8>
v= <8,3>
= 0
5. u = ­2i + j
v = 3i + 6j ­ v = <­2,4>
4. u = 1, 1
4 2
6. u = i v = i + 2j = 0
Find 2 vectors in opposite directions that are orthogonal to vector v.
7.<2,7>
8. <­6,1>
What about the orthogonal vectors of <3,2> and <­3,­5>
6.4 NOTES Homework
p 467: 1­23 odd
47­51 all
59­62 all
}
}
WPF and
show work
WPF for all