Topic 1.3 Extended A - Vector Addition and Subtraction

Topic 1.3 ExtendedA - Vector Addition and Subtraction

Unfortunately, objects move. If they didn't, finding the big three would be a cinch.As it is, most of the things we interact with, in fact, we ourselves, move.

Topic 1.3 Extended A - Vector Addition and Subtraction

And, to make matters worse, we move in more than one dimension.In this section we will look at vectors.

A vector is a quantity that has both magnitude (size) and direction.


The direction of particle moving along a line is given by either a + sign (moving in the positive direction) or a - sign (moving in the negative direction).Thus, if a particle is traveling at 2 ft/s along the x-axis, it is moving in the positive x-direction.If a particle is traveling at -8 ft/s along the y-axis, it is moving in the negative y-direction.However, if a particle is moving in the x-y plane, NOT ALONG EITHER AXIS, its direction cannot be given with a simple sign.Instead, we need an arrow to show its direction. Such an arrow is called a vector.A vector has both magnitude (numerical size), and direction.

Not all physical quantities have a direction. For example, pressure, time, energy, and mass do not have direction. Non-directional quantities are called scalars.

x y

x

y

x

y

vect

or


The simplest vector is called the displacement vector.The displacement vector is obtained by drawing an arrow from a starting point to an ending point.

For example, starting at Milwaukee and ending in Sheboygan would have a displacement vector that looks like this:

Note that the displacement vector (black) does not necessarily show the actual path followed (purple).

A displacement is simply adirected change in position, with disregard to the route taken. dis

pla

cem

ent

vect

oractual

path

Note that the displacement vector traces out the SHORTEST

distance between two points.

FYI: The displacement from Sheboygan to Milwaukee has the same magnitude. What is different about it?


The magnitude (size) of a displacement on the x-y plane is easy to obtain using the Pythagorean theorem or the distance formula:

d = (x2 - x1)2 + (y2 - y1)2 distance formula√

√√√√√

For example, the displacement shown has a magnitude given by

d = (x2 - x1)2 + (y2 - y1)2

= (5 - 1 )2 + (4 - 1 )2

= 42 + 32

= 16 + 9

= 25

= 5 ft.

x (ft)

y (ft)

(x1 , y1) = (1 , 1)

(x2 , y2) = (5 , 4)

start

end


Of course, a scale drawing would allow you to measure the magnitude of the displacement directly with a ruler:

x (ft)

y (ft)

(x1 , y1) = (1 , 1)

(x2 , y2) = (5 , 4)

start

end

Note that the length of the arrow is 5 ft.


In print, vectors are designated in bold non-italicized print.

Consider two vectors: vector a and vector b.

For our purposes, when we are writing a vector symbol on paper, use the letter with an arrow symbol over the top, like this:Each vector has a tail, and a tip (the arrow end).

abtail

tail

tip

tip


Suppose we want to find the sum of the two vectorsa + b = s.We take the first-named vector a, and translate the second-named vector b towards the first vector, SO THAT THE TAIL OF b CONNECTS TO THE TIP OF a.The result of the sum, which we are calling the vector s, is gotten by drawing an arrow from the tail of a to the tip of b.

We are giving the sum an arbitrary

name - say s.

abtail

tail

tip

tip

"translate" means to

move without rotation.

stail

tip


We can think of the sum a + b = s as the directions on a pirate map:

Arrgh, matey. First, pace off the

first vector a.Then, pace

off the second

vector b.And ye'll be

findin' a treasure, aye!


We can think of the sum a + b = s as the directions on a pirate map:We start by pacing off the vector a, and then we end by pacing off the vector b. The treasure is at the ending point.

ab

start

enda+ b = s

s

The vector s represents the shortest distance to the treasure.


Now, suppose we want to subtract vectors.

b

-b

the vector b

the opposite of the vector b

a

-b

a+-b

a-b = a+-bThus,

Just as you learned how to subtract integers by "adding the opposite," so, too, will we subtract vectors.Thus the difference of two vectors a - b is given by

a - b = a + -b. difference of two vectorsWe just have to define the "opposite of b" or "-b."The opposite of a vector is the vector that is the same length, but points in the opposite direction.


Observe the addition of vectors is commutative:a + b = b + a

c

ba(a+

b)

c

ba

(a+b)+c

(b+c)a+(b+c)

Observe that addition of vectors is associative:(a + b) + c = a + (b + c)

b

a

a

b

a+b

b+a


A pirate takes 6 paces east, 4 paces north, 2 paces west, and 1 pace south. (a) Draw a vector diagram showing the pirate's path.

(b) How far does the pirate walk?

(c) Draw in the displacement vector.

(d) What is the magnitude of the displacement?

N (pc)

E (pc)6 paces

4 paces

2 paces

1 pace

6 paces 4 paces 2 paces 1 pace+ + + = 13 paces

13 paces

5 paces


Suppose you know a vector's magnitude and direction.

opphyp

adjhyp

oppadj

hypotenuse

adjacent

opposite

θ

trigonometric ratios

s-o-h-c-a-h-t-o-a

θ

d

dx = d cos θ

dy = d sin θ

d

dxdy

d

d

dx = d cos θ

dy = d sin θ

sin θ = cos θ = tan θ =dx

dy

You can find its components using the trigonometric functions:


Recall the four quadrants of the Cartesian coordinate system:

x(+)x(-)

y(+)

y(-)

III

III IVI + +II - +III - -IV + -

Quadrant x y

Components will "inherit" the signs of the quadrants.

For example, a vector in Quad II will have a negative x-component and a positive y-component.

A vector in Quad III will have negative x-component and a negative y-component.


Standard angles are measured with respect to the +x-axis in a counter-clockwise rotation:

If these angles are used in

dx = d cos θ and dy = d sin θthe correct component signs will be automatic. Reference angles are measured with respect to either x(+) or x(-), whichever is smaller.

If these angles are used in

dx = d cos θ and dy = d sin θthe correct component signs will NOT be automatic.Just use the table on the previous page for the signs of the components.

x(+)x(-)

y(+)

y(-)

θ1

θ2

standard angles

x(+)x(-)

y(+)

y(-)

θ1

θ2

reference angles


45°

Suppose you have a displacementd of 200-m at 135º. Find itscomponents.

Using standard angle:

dx = d cos θ = 200 cos 135° = -141.4 m

dy = d sin θ = 200 sin 135° = 141.4 m SIGNS AUTOMATICUsing reference angle:

dx = d cos θ = 200 cos 45° = 141.4 m

dy = d sin θ = 200 sin 45° = 141.4 m SIGNS NOT AUTOMATIC

dx is (-), and dy is (+) so that

dx = -141.4 m

dy = 141.4 m

x(+)x(-)

y(+)

y(-)

135°

standard angle

x(+)x(-)

y(+)

y(-)

reference angle

180° - 135° = 45°

dx (-)

d y (+

)Since the signs are

not automatic, sketch in components:


Many angles will be given with respect to the points of the compass.

Often a compass is divided into 45° increments.

Then northeast is precisely 45° between north and east.

Sometimes the compass is divided into 22.5° increments.

Then we can speak of NNE, and ENE, and assign a precise angle to each.

ENE

NNE


Finally, we can speak of angles such as "30° east of north," which is a reference to an angle drawn 30° from the north direction, in the eastward direction. 30°

30° east of north

north

Then you can draw your own, exact, right triangle.

30°

60°


Consider the two vectors A and B shown below.

y (ft)

x (ft)

A

B

To add the vectors by components, simply add the x-components, then add the y-components.

Ax = 3 ft

Ay = 2 ft

Bx = -1 ft

By = -2 ft

R = A + BRx = Ax + Bx = 3 ft + -1 ft

= 2 ftRy = Ay + By = 2 ft + -2 ft

= 0 ft

y (ft)

x (ft)

Graphically by components

Rstart end

R

y (ft)

x (ft)start end

AB

R

Graphically


A vector can also be expressed without an angle, and without a picture. This method is often preferred, because it requires no pictures, and therefore less paper. But, you can always make sketches if it helps.To facilitate this method we have to define unit vectors. Unit vectors have UNIT LENGTH (meaning a length of 1) and NO QUANTITY.For example, we could use the directions N, S, E, and W as unit vectors, and express any vector in terms of these unit vectors.Thus the displacement D1 of 30 miles to the north would look like this: D1 = 30 miles N Thus the displacement D2 of 40 miles to the west would look like this: D2 = 40 miles W We could also express D2 in terms of the East unit vector: D2 = -40 miles E

FYI: In fact, for 2D we only need 2 unit vectors, N and E. Then west is -E and south is -N.

Then the sum of the vectors D = D1 + D2 can be written

D = 30 miles N + -40 miles E

^


Naturally, we are not pirates, and so we don't use the point of the compass as unit vectors. Physicists instead use the following:

x is the unit vector in the +x-direction^

y is the unit vector in the +y-direction^

z is the unit vector in the +z-direction^

Unit Vectors

FYI: Some books use i, j, and k for the unit vectors in the x, y, and z-directions.

We read x as "ex hat." (Really...)^

x

y

zWe rarely need 3D, but if we do, here is the standard configuration of the three axes:And here are the unit vectors:

x

y

z

Incidentally, they don't have to start at the origin:

x

y

z


To see how the unit vectors work, let's redo the a previous problem where we found R = A + B:

y (ft)

x (ft)

A

B

A =

R

3x

x

y

^ + 2y (ft)^

B = -1x - 2y (ft)^

R = 2x + 0y (ft)^

+

FYI: Your book uses x instead of x. We will use the x notation in this class because it is the standard.

^ ^

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Topic 1.3 Extended A - Vector Addition and Subtraction