Vectors7.4

JMerrill, 2007

Revised 2009

Definitions• Vectors are quantities that are described by

direction and magnitude (size).

• Example: A force is a vector because in order to describe a force, you must specify the direction in which it acts and its strength.

• Example: The velocity of an airplane is a vector because velocity must be described by direction and speed.

• The vector is the zero vector. It has no direction.

0,0

Representation of Vectors• The velocities of 3 airplanes, two of which are

heading northeast at 700 knots, are represented by u and v

u v We say u = v to indicate both

planes have the same velocity.

w ≠ u or v, why?

The direction is different.

w

Magnitude• The magnitude of vector v is represented by the

absolute value of v.

• In the previous example, |u| = 700, |v|= 700, and |w| = 700. We know that |u|=|v|=|w|, but u ≠ w, and v ≠ w, why?

• The direction is not the same!

Addition of Vectors

AB55555555555555

vA

B

C

AB BC AC

AC is the vector sum

of AB BC

555555555555555555555555555555555555555555

55555555555555

5555555555555555555555555555

10

5

Addition of Vectors• Addition is commutative, so a + b = b + a

• The vector sum is called the resultant.

a

b

a+bb

a

b+a

Vector Subtraction• A negative vector has the same magnitude, but

in the opposite direction.

• v + (-v) = 0

• v – w means v + (-w)

v

v

w

-v

Multiples of Vectors

v

2v

3v

-v

-2v

-3v

You Do• Let u = and v =

• Find 3u + 2v

• Find ½ u + 4v

• Find u – 2v

Scalar Multiplication• Real numbers are often referred to as scalars.

• When we multiply a vector by a scalar, we use the same rules that we are familiar with:

k(v + w) = kv + kw

k(mv) = kmv

Component Form• From the tail to the tip of

vector v, we see:

• A 2 unit change in the x-direction, and

• A -3 unit change in the y-direction.

• 2 and -3 are the components of v.

• When we write v = we are expressing v in component form.

2

3

2, -3 ,

Component Form• You can count the number

of spaces to get the component form or, you can subtract the coordinates.

• IT IS ALWAYS B – A!

• The magnitude of vector AB is found using the distance formula:

(x1,y1

)

(x2,y2

)

(x2 – x1)

(y2 – y1)2 1 2 1, 55555555555555AB x x y y

2 22 1 2 1( ) ( )

55555555555555AB x x y y

Example• Given A(4, 2) and B(9, -1), express in

component form. Find 55555555555555AB

9 4, 1 2 5, 3 55555555555555AB

2 25 ( 3) 34 55555555555555AB

AB55555555555555

Vector Operations with Coordinates Vector Addition

v + u =

Vector Subtractionv - u =

Scalar Multiplicationkv =

a,b + c,d = a+c, b+d

a,b - c,d = a-c, b-d

k a,b =ka, kb

Example• If u = and v = , find:

• u + v

• u – v

• 2u – 3v

1, 3 2,5

1+2, -3+5 = 3, 2

1 - 2, -3 - 5 = -1, -8

2 1, -3 - 3 2, 5 = 2, -6 - 6, 15 = -4, -21

Drawing• Draw a parallelogram if you have a force.

• Draw using tip-to-tail if you have a change of course.

• ALWAYS, ALWAYS, ALWAYS make your drawing in proportion.

• And remember, heading/bearing/compass direction is always measured clockwise from magnetic north!

Example• A force of 20N (20 Newtons) is pulling an object

east and another force of 10N is pulling the object in the compass direction of 150o. Find the magnitude and direction of the resultant force.

• Let O = object (and make it at the origin for ease of computation)

Example

10N

1. Draw what you know

2. Draw the parallelogram

3. Draw the resultant—that’s what you’re looking for!

4. Use Law of Sines/Cosines to find magnitude and direction. We will use only one of the triangles.

150o

20N

60o

x

x

Example

10N

1. We know the obtuse angle = 120 degrees.

2. Let r = resultant. Use the Law of Cosines to find r.

3. Now find angle x:

20N

120o 2 2 210 20 2(10)(20)cos120

26.46

or

r

sin sin12010 26.46

sin 0.3273

19.1

o

o

x

x

x

90o + 19.1o = 109.1o

So the resultant force is 26.46N in a direction of 109.1o

x

You Do:• An airplane has a velocity of 400mph southwest.

A 50mph wind is blowing from the west. Find the resultant speed and direction of the plane.

• We’re changing direction, so use tip-to-tail.

• The plane’s resultant velocity is about 366mph on a course of approximately 219.5 degrees