c hapter 6 6-4 vectors and dot products. o bjectives find the dot product of two vectors and use the...
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CHAPTER 6 6-4 Vectors and dot products
OBJECTIVES
Find the dot product of two vectors and use the properties
Find the angle between two vectors Write vectors as the sum of two vectors
components Use vectors to find the work done by a force
DOT PRODUCT OF TWO VECTORS
Dot Product is a third vector operation. This vector operation yields a scalar (a single number) not another vector. The dot product can be positive, zero or negative.
Definition of dot product
2121 bbaa wvjiwjiv 2211 and where baba
EXAMPLE
find ,4 and 52 If wvjiwjiv
1542 wv 8 5 3
This is called the dot product. Notice the answer is just a number NOT a vector.
EXAMPLE #1: FINDING DOT PRODUCTS
Find each dot product.
A) <4, 5>●<2, 3> sol: 23
B) <2, -1>●<1, 2> sol:0
C) <0, 3>●<4, -2> sol:-6
D) <6, 3>●<2, -4> sol: 0
E) (5i + j)●(3i – j) sol:14
PROPERTIES OF DOT PRODUCTS
Let u, v, and w be vectors in the plane or in space and let c be a scalar.1. u●v = v●u2. 0●v = 03. u●(v + w) = u●v + u●w4. v●v = ||v||2
5. c(u●v) = cu●v = u●cv
EXAMPLE#2 USING PROPERTIES OF DOT PRODUCTS
Let u=<-1,3>, v=<2,-4> and w=<1,-2>. Use vectors and their properties to find the indicated quantity
A. (u.v)w sol: <-14,28> B.u.2v sol: -28 C.||u|| sol:
CHECK IT OUT!
Let u = <1, 2>, v = <-2, 4> and w = <-1, -2>. Find the dot product.
A) (u●v)w
B) u●2v
CHECK IT OUT !
Given the vectors u = 8i + 8j and v = —10i + 11j find the following.
A. B.
vuvv
ANGLE BETWEEN TWO VECTORS
Angle between two vectors (θ is the smallest non-negative angle between the two vectors)
wvwv
cos
wvwv1cos
EXAMPLE
Find the angle between
2,3,5,1 vu
SOLUTION
2222 23,)5(1
132531
2,3,5,1
vu
vu
vu
vu
vuCos
SOLUTION
457071.0
...7071.0
338
13
1326
13
23,)5(1
132531
2,3,5,1
1
2222
Cos
Cos
Cos
vu
vu
vu
CHECK IT OUT!
Find the angle between u = <4, 3> and v = <3, 5>.
Solution: 22.2 degrees
DEFINITIONS OF ORTHOGONAL VECTORS
The vectors u and v are orthogonal if u●v = 0.
Orthogonal = Perpendicular = Meeting at 90°
EXAMPLE: ORTHOGONAL VECTORS
Are the vectors u = <2, -3> and v = <6, 4> orthogonal?
EXAMPLE
Determine if the pair of vectors is orthogonal
jivjiu316
2and38
jivjiu 921and37
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 Find the projection of u into v. Then write u
as the sum of two orthogonal vectors Sol:
vv
vuuprojw
wwu
vu
v
21
21
2,6,5,3
SOLUTION
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
WORK
The work W done by a constant force F in moving an object from A to B is defined as
ABW FThis means the force is in some direction given by the vector F but the line of motion of the object is along a vector from A to B
DEFINITION OF WORK
Work is force times distance.If Force is a constant and not at an angle
If Force is at an angle
PQFprojWPQ
EXAMPLE
To close a barn’s sliding door, a person pulls on a rope with a constant force of 50 lbs. at a constant angle of 60 degrees. Find the work done in moving the door 12 feet to its closed position.
Sol: 300 lbs
EXAMPLE
Find the work done by a force of 50 kilograms acting in the direction 3i + j in moving an object 20 metres from (0, 0) to (20, 0).
SOLUTION
Let's find a unit vector in the direction 3i + j
10133 22 ji
jiu10
1
10
3
Our force vector is in this direction but has a magnitude of 50 so we'll multiply our unit vector by 50.
jiF10
1
10
350
SOLUTION
jiji 02010
1
10
350
W
kgs m 948.68 10
3000W
STUDENT GUIDED PRACTICE
Do7,9,11,23,31,43 and 57 in your book page 440 and 441
HOMEWORK
Do 8,10,12,14,24,32,44 and 69 in your book page 440 and 441
CLOSURE
Today we learned about vectors and work Next class we are going to learn about
trigonometric form of a complex number