Dot Product

The dot product of and 1 2,v v v

is 1 1 2 2u v u v u v

1 2,u u u

Projection of a Vector and Vector Components When we want a

component of a vector along a particular direction, it is useful to think of it as a projection.

The projection always has length a cos , where a is the length of the vector and is the angle between the vector and the direction along which you want the component.

You should know how to write a vector in unit vector notation

cosa

cosa

sina

ˆ ˆ or ,x y x ya a i a j a a a

Dot Product The dot product says

something about how parallel two vectors are.

The dot product (scalar product) of two vectors can be thought of as the projection of one onto the direction of the other.

Components

AxAAiA

ABBA

cosˆ

cos

B

zzyyxx BABABABA

BA )cos(

)cos( BA

A

B

Projection is zero

Projection of a Vector: Dot Product The dot product says

something about how parallel two vectors are.

The dot product (scalar product) of two vectors can be thought of as the projection of one onto the direction of the other.

Components A

B

zzyyxx BABABABA

Projection is zero

xAAiA

ABBA

cosˆ

cos

Derivation How do we show that ? Start with

Then

But

So

zzyyxx BABABABA

kBjBiBB

kAjAiAA

zyx

zyx

ˆˆˆ

ˆˆˆ

)ˆˆˆ(ˆ)ˆˆˆ(ˆ)ˆˆˆ(ˆ

)ˆˆˆ()ˆˆˆ(

kBjBiBkAkBjBiBjAkBjBiBiA

kBjBiBkAjAiABA

zyxzzyxyzyxx

zyxzyx

1ˆˆ ;1ˆˆ ;1ˆˆ

0ˆˆ ;0ˆˆ ;0ˆˆ

kkjjii

kjkiji

zzyyxx

zzyyxx

BABABA

kBkAjBjAiBiABA

ˆˆˆˆˆˆ

Vector operations (cont.)

The dot product (scalar product):

The result of the dot product is a scalar:

A·B = |A||B| cos = AB cos

A·B = maximum for = 0A·B = minimum for = A·B = zero for AB

The dot product is commutative:

A·B = B·A

B

A The dot product is distributive:

A· (B+C)=A·B+A·C

Vector Components (cont.)

To calculate the dot product of two vectors, multiply like components and add:

A·B = AxBx + AyBy + AzBz

To calculate the cross product of two vectors, evaluate the following determinant:

AB

ˆ x ˆ y ˆ z

Ax Ay Az

Bx By Bz

8

Definitions Quantities such as length area, volume, temperature, and time have

magnitude only and are completely characterized by a single real number with appropriate units (such as feet, degrees, or hours).

Such quantities are called scalar quantities, and the corresponding real numbers are scalars.

Quantities that involve both a magnitude and a direction, such as velocity, acceleration, and force, are vector quantities, and they can be represented by directed line segments.

These directed line segments are called vectors. The length of the vector represents the magnitude of the vector

quantity. The direction is indicated by the position of the vector and the

arrowhead at one end.We use the notation AB to name a line segment with endpoints A and B

and AB to name a ray with initial point A and passing through B.

The notation AB is used to name a vector with initial point A and terminal point B. The vector AB terminates at B, while the ray AB goes beyond B.

Two vectors are equal if they have the same magnitude and the same direction.

7.3 Vectors

9

Definition: Scalar MultiplicationFor any scalar k and vector A, kA is a vector with magnitude | k |

times the magnitude of A. • If k > 0, then the direction of kA is the same as the direction of

A.• If k < 0, then the direction of kA is opposite to the direction of

A.• If k = 0, then kA = 0 .

7.3 Vectors

10

Two forces are represented by vectors A and B. If A and B have the same direction, then there would be

a total force A + B. One force A + B acting along the diagonal of the

parallelogram with magnitude equal to the length of the diagonal, has the same effect as the two forces A and B.

In physics, this result is known as the parallelogram law.

The single force A + B acting along the diagonal is called the sum or resultant of A and B.

7.3 Vectors

B

A

A+B

11

Note that the vector A + B coincides with the diagonal of a parallelogram whose adjacent sides are A and B.

A + B =B + A, and vector addition is commutative.If A and B have the same direction or opposite direction, then no

parallelogram is formed. Each vector in the sum A + B is called a component of the

sum. For every A there is a vector – A, the opposite of A, having

the same magnitude of A but in the opposite direction. The sum of a vector and its opposite is the zero vector, A + (–

A) = 0. For any two vectors A and B, A – B = A + (– B).Remember that in any parallelogram the opposite sides are equal

and parallel, and adjacent angles are supplementary.The diagonals of a parallelogram do not bisect the angles of a

parallelogram unless the adjacent sides of the parallelogram are equal in length.

7.3 Vectors

12

Horizontal and Vertical Components Any nonzero vector w is the sum of a vertical component and a

horizontal component. The horizontal component is denoted wx and the vertical component

is denoted wy . The vector w is the diagonal of the rectangle formed by the vertical

and horizontal components. If a vector w is placed in a rectangular coordinate system so that its

initial point is the origin, then w is called a position vector or radius vector.

The angle (0o < 360o) formed by the positive x-axis and a position vector is the direction angle for the position vector (or any other vector that is equal to the position vector).

If the vector w has magnitude r , direction angle , horizontal component wx and vertical component wy , then by using trigonometric ratios we get

If the direction of w is such that sin or cos is negative, then we can write

| wx | = | r cos | and | wy | = | r sin |.

7.3 Vectors

.sin and cosor sin and cos rwrwr

w

r

wyx

yx

13

Component Form of a VectorAny vector is the resultant of its horizontal and vertical components.

Since the horizontal and vertical components of a vector determine the vector, it is convenient to use a notation for vectors that involves them.

The notation is used for the position vector with terminal point (a, b). The form is called component form because its horizontal

component is and its vertical component is Since the vector extends from (0, 0) to (a, b), its magnitude is

the distance between these points:

When vectors are written in component form , operations with vectors are easier to perform.

The endpoint of A + B is (a1 + b1, a2 + b2) and so

The sum can be found in component form by adding the components instead of drawing directed line segments.

7.3 Vectors

ba,

0,aba,

.,0 bbav ,

22 bav

., 2211 baba BA

2121 , and , bbaa BA

14

Rules for Scalar Product, Vector Sum, Vector Difference, and Dot Product

If and k is a scalar, then Scalar product Vector sum Vector

difference Dot product

7.3 Vectors

,,,, 2121 bbaa BA

21,kakak A

2211 , baba BA

2211 , baba BA

2211 baba BA

15

The Angle Between Two Vectors If A = kB for a nonzero scalar k, then A and B are

parallel vectors. If A and B have the same direction (k > 0) the angle

between A and B is 0o. If they have opposite directions (k < 0) the angle

between them is 180o. If A and B are non-parallel vectors with the same initial

point, then the vectors A, B, and A – B form a triangle. The angle between the vectors A and B is the angle

. The angle between two vectors is in the interval [0o,

180o]. If the angle between A and B is 90o, then the vectors

are perpendicular or orthogonal.

7.3 Vectors

16

Theorem: Dot ProductIf A and B are nonzero vectors and is the angle between

them, then

7.3 Vectors

.cosBABA

17

Proof: The vectors, A, B, and A – B form a triangle.Apply the law of cosines to this triangle and simplify as

follows.

Note that cos = 0 if and only if A B = 0. So two vectors are perpendicular if and only if their dot product is zero.

Two vectors are parallel if and only if

7.3 Vectors

.1cos

BABA

BABA

BA

BA

BA

BABABA

cos

cos

cos

22cos2

cos2

cos2

2211

2211

222

211

22

21

22

21

222

baba

baba

bababbaa

18

Unit Vectors The vectors are called unit vectors because each has

magnitude one. For any vector we have

The form a1i + a2j is called a linear combination of vectors i and j.

These unit vectors are thought of as fundamental vectors, because any vector can be expressed as a linear combination of them.

7.3 Vectors

1,0 and 0,1 ji21, aa

ji 212121 1,00,1, aaaaaa

a2 j

a1 i

a1 i + a2 j

x

y

19

We can use the vector v1 to represent the heading and air speed of a plane.

The vector v2 represents the wind direction and speed. The resultant of v1 and v2 is the vector v3, where v3 represents the

course and ground speed of the plane. The angle between the heading and the course is the drift angle. Recall that the bearing of a vector used to describe direction in air

navigation is a non-negative angle smaller than 360o measured in a clockwise direction from due north.

7.3 Vectors

S

E

N

W v1

v2 v3

Drift angle