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Coordinate Systems & Vectors

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Coordinate Systems and Frames of Reference

The location of a point on a line can be described by one coordinate; a point on a plane can be described by two coordinates; a point in a three dimensional volume can be described by three coordinates. In general, the number of coordinates equals the number of dimensions. A coordinate system consists of:

1. a fixed reference point (origin) 2. a set of axes with specified directions and scales

3. instructions that specify how to label a point in space relative to the origin and axes

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Coordinate Systems

In 1 dimension, only 1 kind of system, Linear Coordinates (x) +/-

In 2 dimensions there are two commonly used systems, Cartesian Coordinates (x,y) Polar Coordinates (r,θ)

In 3 dimensions there are three commonly used systems,

Cartesian Coordinates (x,y,z) Cylindrical Coordinates (r,θ,z) Spherical Coordinates (r,θ,φ)

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Cartesian coordinate system also called rectangular

coordinate system x and y axes points are labeled (x,y)

Plane polar coordinate system

origin and reference line are noted

point is distance r from the origin in the direction of angle θ

points are labeled (r,θ)

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The relation between coordinates

x rcosθ= θ= sinry

22 yxr +=

xytan =θ

Furthermore, it follows that

Problem: A point is located in polar coordinate system by the coordinate and .

Find the x and y coordinates of this point, assuming the two coordinate systems have the same origin.

5.2r =35=θ

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Example : The Cartesian coordinates of a point are given by (x,y)= (-3.5,-2.5) meter. Find the polar coordinate of this point.

Solution:

21636180 =+=θ

714.05.35.2

xytan =

−−

==θ

m3.4)5.2()5.3(yxr 2222 =−+−=+=

Note that you must use the signs of x and y to find that is in the third quadrant of coordinate system. That is not 36

θ216=θ

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Scalars and Vectors Scalars have magnitude only. Length, time, mass, speed and volume are examples of scalars . Vectors have magnitude and direction .The magnitude of is written Position, displacement, velocity, acceleration and force are examples of vector quantities.

vv

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Properties of Vectors

Equality of Two Vectors Two vectors are equal if they have the same magnitude and the same direction

Movement of vectors in a diagram Any vector can be moved parallel to itself without being affected

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Negative Vectors Two vectors are negative if they have the same magnitude but are 180° apart (opposite directions)

Multiplication or division of a vector by a scalar results in a vector for which (a) only the magnitude changes if the scalar is positive (b) the magnitude changes and the direction is reversed if the scalar is negative.

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Adding Vectors

When adding vectors, their directions must be taken into account and units must be the same

First: Graphical Methods

Second: Algebraic Methods

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Adding Vectors Graphically (Triangle Method)

Continue drawing the vectors “tip-to-tail”

The resultant is drawn from the origin of A to the end of the last vector

Measure the length of R and its angle

General physics I, lec 2 By: T.A.Eleyan 12

When you have many vectors, just keep repeating the process until all are included

The resultant is still drawn from the origin of the first vector to the end of the last vector

General physics I, lec 2 By: T.A.Eleyan 13

Alternative Graphical Method (Parallelogram Method)

When you have only two vectors, you may use the Parallelogram Method All vectors, including the resultant, are drawn from a common origin

The remaining sides of the parallelogram are sketched to determine the diagonal, R

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Vector Subtraction

Special case of vector addition If A – B, then use A+(-B) Continue with standard vector addition procedure

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Components of a Vector

These are the projections of the vector along the x- and y-axes

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The x-component of a vector is the projection along the x-axis The y-component of a vector is the projection along the y-axis

Then,

cosxA A θ=

sinyA A θ=

x yA A= +A

x

y12y

2x A

AtanandAAA −=θ+=

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Adding Vectors Algebraically

(1)Choose a coordinate system and sketch the vectors (2)Find the x- and y-components of all the vector (3)Add all the x-components

This gives Rx:

∑= xx vR

∑= yy vR

(4)Add all the y-components

This gives Ry

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(5)find the magnitude of the Resultant

Use the inverse tangent function to find the direction of R:

2y

2x RRR +=

x

y1

RR

tan−=θ

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Unit Vectors

A Unit Vector is a vector having length 1 and no units

It is used to specify a direction. Unit vector u points in the direction of

U Often denoted with a “hat”: u = û

U = |U| û

û

x

y

z i

j

k

Useful examples are the cartesian

unit vectors [ i, j, k ] Point in the direction of the

x, y and z axes. R = rx i + ry j + rz k

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Example : A particle undergoes three consecutive displacements given by

,cm)kj3i(d1 −+= cm)k3ji2(d 2 −−= cm)ji(d3 +−=

Find the resultant displacement of the particle

cm)k4j3i2(Rk)031(j)113(i)121(dddR 321

−+=+−−++−+−+=++=

cm4R,cm3R,cm2R zyx −===

cm39.5RRRR z2

y2

x2 =++=

Solution:

The resultant displacement has component

The magnitude is

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Product of a vector

1-The scalar product (dot product )

There are two different ways in which we can usefully define the multiplication of two vectors

Each of the lengths |A| and |B| is a number and is number, so A.B is not a vector but a number or scalar. This is why it's called the scalar product.

Special cases of the dot product Since i and j and k are all one unit in length and they are all mutually perpendicular, we have

i.i = j.j = k.k = 1 and i.j = j.i = i.k = k.i = j.k = k.j = 0.

θcos. BABA =

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The angle between the two vector If A and B both have x,y and z components, we express them in the form

kAjAiAA zyx ++= kBjBiBB zyx ++=

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2- The vector product (cross product)

Special cases of the cross product

nABBA ˆsin. θ=

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Problem 1: Find the sum of two vectors A and B lying in the xy plane and given by

Problem 2: A particle undergoes three consecutive displacements :

Find the components of the resultant displacement and its magnitude.

mjiBmjiA )42(,)22( −=+=

cmjid

cmkjidcmkjid

)1513(

,)51423(,)123015(

3

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+−=

−−=−+=