Introduction to Vectors

Why Vectors? • Say you wanted to tell your friend that you’re running late

and will be there in five minutes. That’s precisely enough information for your friend to know when you’ll arrive – Time is a scalar quantity.

• Now suppose that on your way to your friend’s you forget you needed something from the mall and phone your friend telling him you’ll meet him 2 miles from the mall to save time. Is this enough information?

Why Vectors • Some physical quantities, like position, require more than just

a magnitude (number amount with units). • Some physical quantities, called vectors, require we specify

direction too.

mall

2 mi

Additional information is required to know where your friend is

Vector quantities are denoted in the text and notes by placing an arrow over the symbol representing the physical quantity. Sometimes, instead of placing an arrow above it, the symbol is simply BOLDFACED.

or The magnitude of a vector V is expressed as:

or It is the numerical value of the vector inclusive of any units

and absent of direction.

V

V V

V

(1a) (1b)

How are Vectors Represented?

(2a) (2b)

Given the vector, x = 12.4 meters due south, the magnitude of x, or | x |, is 12.4 meters

How are Vectors Represented? Like most physical quantities, the symbol used to represent a

particular vector is usually derived from the first letter in the name of that quantity.

Physical Quantity Vector Symbol velocity v or v

acceleration a or a force F or F

momentum p or p

I did say

usually!

How are Vectors Represented? In addition to representing vector quantities symbolically, a

vector can also be depicted graphically.

This is done by drawing an arrow whose length is representative of the physical length of the quantity being expressed.

|n| 2|n|

Vectors are drawn from their point of origin (tail) to their endpoint (arrow head). The vector n above has some actual length that is a scaled reconstruction of the real physical length of the vector quantity it is intended to represent.

2|n|

How are Vectors Represented? Notice that the middle image is twice as long as the image on

the left and therefore has twice the magnitude. 2|n| versus |n|, and a measurement of these would confirm this.

Notice too that the image to the right is twice as long as that of the image on the left and its magnitude is likewise twice as great even though the vector is oriented differently.

Adding (Subtracting) Vectors Like scalar quantities, (eg. counting numbers), vectors

quantities can be added and subtracted from other vector quantities provided the vectors represent the same physical entity.

For example, one can add one velocity vector to another velocity vector but not to an acceleration vector. Similarly, one can subtract one force vector from another but not a force from a momentum. (Can only add to )

Steps for Adding Vectors Graphically To add two vectors graphically:

1.) Start by picking an appropriate scale. Since most graph paper is 8 ½ inches x 11 inches, the very largest one can make something is 11 inches (~0.28 m). So if you are trying to represent the velocities of two objects that are on the order of 40 m/s then you might want to make 80 m/s equal 0.28 m or the whole length of the page, resulting in a scale factor of 80 m/s : 0.28 m.

2.) Next, establish a coordinate system – usually a Cartesian grid and label the principal axes (x,y; E,N; right, up)

3.) Starting from the origin, construct your first vector by orienting it correctly with respect to the grid and scaling it appropriately.

Steps for Adding Vectors Graphically

4.) At the tip (arrowhead) of the first vector, construct a second coordinate axes aligned to the first.

5.) Construct the second vector so it too is oriented correctly and draw it the correct scaled length, starting once again from the origin.

6.) Translate the second vector so that its tail aligns with the arrowhead of the first vector. This is the head to tail method

7.) Construct a vector from the base of the first (tail) vector to the tip (arrowhead) of the second. This is what is referred to as the Resultant Vector, (result of adding).

Adding Vectors Graphically (Example) Let’s use an example to work through the technique! A boat is traveling at 4 m/s due north across a river that has a

current of 3 m/s directed to the east. What is the magnitude of the boats velocity relative to the ground?

STEP 1. Set Scale STEP 2. Set Coordinates one grid mark = 2 m/s

2 m/s

N

E

Adding Vectors Graphically (Example) STEP 3. Construct 1st vector STEP 4. Construct 2nd axes 4 m/s N = 2 grids N at tip of 1st vector

N

E

N

E

v1 v1

Adding Vectors Graphically (Example) STEP 5. Construct 2nd vector STEP 6. Translate 2nd vector 3 m/s E = 1.5 grids E (no stretching & no rotating)

N

E

v1

v2

v1

N

E v2

v1

v2

Adding Vectors Graphically (Example) STEP 7. Construct resultant STEP 8. Measure resultant

N

E v2

v1

v2

N

E v2

v1

v2

+v2 v1

13 m/s – 8 m/s = 5 m/s

Adding (Subtracting) Vectors While graphical techniques can lend insight into the problem

being worked, and can lead to an approximate solution, they often are prone to construction and scaling errors.

A more accurate and less labor intensive technique involves an analytical (math based) approach.

Let’s begin with the fact that the sum of two vectors, A and B,

can be expressed algebraically as: S = A + B, provided that the vectors both represent the same physical quantity.

Adding (Subtracting) Vectors When adding vectors analytically, we are restricted to only

being able to add vectors that are perpendicular to each other or vectors that are parallel to one another.

This would seem overly limited except for the fact all vectors can be resolved into parallel and perpendicular components.

As an example, consider: We can rewrite A and B as the sums of their respective x, y,

and z components:

BAS += (3)

(4b) zyx aaaA ++= zyx bbbB ++=(4a)

Adding (Subtracting) Vectors

Notice, that in resolving each vector, we have made use of

the scalar multiplication property that an = |an|u, where |an|is the magnitude of the vector an. The direction of the vector is along the direction of unit vector u.

In this case, the coefficients ax, bx , ay , by , az and bz are the scalar components along each of the three principle spatial dimensions designated by the i, j, and k unit vectors. Since these unit vectors each have a magnitude of one, they do not further scale the coefficients, they only provide direction.

kbjbibkbjbibB zyxzyx ++=⋅+⋅+⋅= (5b)

kajaiakajaiaA zyxzyx ++=⋅+⋅+⋅= (5a)

Adding (Subtracting) Vectors The sum of vectors A and B is then:

( ) ( ) ( ) ( ) ( ) ( )kbajbaibakbajbaibaS zzyyxxzzyyxx +++++=⋅++⋅++⋅+=

(6)

( ) ( ) ( )kbajbaibaSSSS zzyyxxzyx +++++=++= (7)

Sx Sy Sz

Sx Sy Sz

Notice that we have gone from two arbitrarily directed vectors, A and B which we could not add analytically to three vector components Sx, Sy, and Sz , that are mutually perpendicular, and which can be added.

Adding (Subtracting) Vectors In equations 6 and 7, the parallel vector components add like

any real number. For example if ax = 3 and bx = -1 , then ax + bx = 2. Similarly, if ay = 2 and by = 0, then ay + by = 2.

Perpendicular components add in a way analogous to that described by the Pythagorean theorem, a way known as quadrature.

Given the example above, where Sx = (3+-1)i, and Sy = (2 + 0)j, Sx = 2i is perpendicular to Sy= 2j . To find the magnitude of

the sum of these vectors, we must first square them, add, and then take the square root as shown below:

( ) ( ) 2222 222222 =+=+++=+=+= yyxxyxyx babaSSSSS(8)

Why Use Pythagorean Addition? Let’s look at the problem again, graphically.

y

x Sx

Sy

Sx

+Sy Sx

Notice first that the first vector, Sx, the one in the x-direction, is orthogonal, that is perpendicular, to the second vector that has only a y-component. They each form one of the two legs of a right triangle. The hypotenuse, which is the sum of the two vectors, |Sx+ Sy|, is found using Pythagoras:

22 22 +=c

If given the coordinate location of the end points of vectors,

you can simply add the coordinate values of each vector’s endpoint to one another to arrive at the coordinate location of the resultant vector’s end point. This assumes that the initial point of all the vectors is the origin.

For example, if A = <ax,ay> = <3,4> and B = <bx,by> = <-2,5>,

then A + B = = <ax+bx, ay +by> = <1,9>. Other combinations are possible. For example: 4 A – 2 B = <16,6>.

Adding (Subtracting) Vectors

Adding Vectors Analytically (Example) Let’s use the same example we used to demonstrate the

graphical approach to work through this technique again! A boat is traveling at 4 m/s due north across a river that has a

current of 3 m/s directed to the east. What is the magnitude of the boats velocity relative to the ground? north = j or y

kjikvjvivv zyx 0401111 ++=⋅+⋅+⋅=

kjikvjvivv zyx 0032222 ++=⋅+⋅+⋅= (9b)

(9a)

( ) ( )kjikjivv 00304021 +++++=+

( )kjivv 04321 ++=+

m/s 5043 22221 =++=+ vv

(10)

(11)

(12)

Adding Vectors Analytically (Example) So the magnitude of the resultant vector is 5 m/s, what about

it’s direction? The direction is found using trigonometry. Looking again at the graphic, we see that the resultant makes an angle θ with respect to the positive x-axis.

xx

yy

vvvv

21

211tan++

= −θN

E v2

v1

v2

+v2 v1 θ

°=++

= − 13.530430

tan 1θ

Sum of all y-values

Sum of all x-values

Vectors can be added together by graphical means by translating the second vector so that its tail intersects with the head of the first as shown below.

V1

V2

V1

V2

V1 + V2

Care must be taken not to rotate or stretch the vector when translating.

Vectors - Summary

When representing vector quantities graphically, it is important to maintain the same relative scale.

Two vectors can be added using the Parallelogram Law

u

v

u + v

Analytically, vectors can be added only if they are parallel or perpendicular to each other.

Vectors - Summary

Special Unit Vectors

A vector such as <3,4> can be written as 3<1,0> + 4<0,1>. For this reason, these vectors are given special names: i = <1,0> and j = <0,1> or unit vectors. A vector in component form V = <m,n> can be written mi + nj.