12.2 Vectors

Quantities that have magnitude but not direction are called scalars.

Ex: Area, volume, temperature, time, etc. Quantities such as force, acceleration, velocity or

displacement that have direction as well as magnitude are represented by directed line segments, called vectors.

A

B

initialpoint

terminalpointAB

The length of the vector is calledthe magnitude and is denoted by

AB

Definitions

A vector is in standard position if the initial point is at the origin.

x

y

1 2,v v The component form of this vector

is:1 2,v vv

Vectors are equivalent if they have the same length and direction (same slope).

then the component form of

PQ

is:

( , ) ( , )P c d and Q a b

,a c b d

If

are initial and terminal points of a vector, P

Q

(c,d)

(a,b)

v(a-c, b-d)

x

P

Q

(-3,4)

(-5,2)

The component form of

PQ

is: 2, 2 v

v(-2,-2) 2 22 2 v

8

2 2

The magnitude is

Example

i

and1,0i

j

If 0v then v is a zero vector : 0 0,0

are called the standard basis vectors.

0,1j

The magnitude of 1 2,v vv is: 2 21 2v v v

If 1v then v is a unit vector.

and

1,0,0i

If 0v then v is a zero vector : 0 0,0,0

are called the standard basis vectors.

0,1,0j

The magnitude of 1 2 3, ,v v vv is:

2 2 21 2 3v v v v

If 1v then v is a unit vector.

0,0,1k

Vectors in Space

1 2 3 1 2 3Let , , , , , , a scalar.u u u v v v k u v

1 1 2 2 3 3, ,u v u v u v u vVector sum:

1 1 2 2 3 3, ,u v u v u v u vVector difference

Scalar Multiplication: 1 2 3, ,k ku ku kuu

Negative (opposite): 1 2 31 , ,u u u u u

Vector v is parallel to u if and only if v = ku for some k.

Vector Operations

v

vu

u

u+vu + v is the resultant vector.

v

vu

u

u-v u - v is the resultant vector.

Parallelogram Law

A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they?

N

E

Application

A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they?

N

Eu

A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they?

N

E

v

u60o

A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they?

N

E

v

u

We need to find the magnitude and direction of the resultant vector u + v.

u+v

N

E

v

u

The component forms of u and v are:

u+v

500,0u

70cos60 ,70sin 60v

500

7035,35 3v

Therefore: 535,35 3 u v

538.4 22535 35 3 u v

and: 1 35 3tan535

6.5

The new ground speed of the airplane is about 538.4 mph, and its new direction is about 6.5o north of east.

ij

1 2v v v i jv is called a linear combination of i and j

Any vectors can be written uniquely in terms of standard basis vectors :

v

cos sin v v i v j

(measured counterclockwise) with the positive x-axis

then v can be written as

If v is any nonzero vector that makes an angle

Linear Combination

1 2 3v v v v i j k

v is called a linear combination of i, j and k

Standard basis vector notation

Linear Combination in Space

1) Find the unit vector in the direction of v

6,0,8v

(4, 2,7), ( 2,0,3), (7, 3,9)

2) Determine whether the points are collinear:

(1,1,3), (9, 1, 2), (11,2, 9), (3,4, 4)

3) Show that the following points form the vertices of a parallelogram:

Examples