I.  WHAT IS A VECTOR?

•  A VECTOR is a variable quantity consisting of two components: o  MAGNITUDE: How big? This can represent length, pressure, rate, and other quantities o  DIRECTION: Which way is the magnitude pointed or exerted?

•  A vector is represented symbolically with an arrow, or in equations as a letter with an arrow over it,

A. MAGNITUDE •  Let’s start thinking about vectors by thinking about the displacement of a point from its origin. The magnitude of the DISPLACEMENT of a point tells how far (length) it is from a starting point or origin. In physics, this is NOT the same as DISTANCE.

UNIT XX: VECTORS

1-dimensional:

a

A

0 1 2 0 1 2

point on a line: x = 2 Above is a vector drawing that represents the displacement of the point from zero

How would you describe the magnitude of this vector?

origin

1

2-dimensional

x

y

point in a plane: x = 3 , y = -2

x

y

vector that describes the displacement of point (3, -2) from the origin

How would you define the magnitude of this vector?

origin

2

•  It is easy to see that the magnitude of the vector above is 2 units.

•  1-dimensional vectors will fall parallel to either the x-axis (horizontal) or the y-axis (vertical).

•  In a 2-dimensional plane, the vector length must account for displacement along TWO axes. We say that the vector has BOTH an x-component and a y-component.

•  You you like, you can think of the 1-dimenional vectors as 2-dimensional vectors that have either an x-component = 0 (vertical) or y-component = 0 (horizontal)

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•  The magnitude of the vector above can be found by measuring the length from the origin to the point (for example, using a ruler or graph paper), or by using the distance formula:

(x2 − x1)2 + (y2 − y1)

2 = (3− 0)2 + (−2 − 0)2 = 9 + 4 = 13 units

x

y vector with y-component = 0

vector with x-component = 0

x vector with x and y components

y

All the x and y-component stuff means is that we can say “how big” a vector is in the x direction and in the y direction. This will be important again when we discuss FORCES !

4

the length of this dotted line is the magnitude of the x-component of the vector

Note the dotted line is parallel to an imaginary x-axis

y

x the length of this dotted line is the magnitude of the y-component of the vector

Note the dotted line is parallel to an imaginary y-axis

The x and y components of a 2-dimensional vector

A

(x1, y1)

(x2, y2)

Ax

Ay

Ax = x2 – x1

Ay = y2 – y1

A = Ax2 + Ay

2

RESOLVING THE X AND Y COMPONENTS OF A VECTOR ALWAYS YIELDS A RIGHT TRIANGLE. THEREFORE, WE CAN USE PYTHAGORAS THEOREM TO COMPUTE THE MAGNITUDE OF THE VECTOR.

5


B.
DIRECTION

•  Direction can be described in a number of different ways. We could indicate a + or – direction, we could use compass directions like east, north-west, etc., we could even say “up and to the right”.

•  For 2-dimensional vectors, it is common to define an angle with reference to the x-axis (East) in an x-y plane. The angle of a vector is defined as the angle formed where the vector and an imaginary x-axis meet, measured in a counter-clockwise direction.

•  Examples: y


x


Angles
of
0o
or
1800

are
NOT
the
same
since
the
direc;on
is
opposite


Also,
angles
of
900
and
270o

are
not
the
same.


vector

angle
=
350


vector
angle
=
1500


REMEMBER:
A
VECTOR
IS
DESCRIBED
BY
BOTH
MAGNITUDE
AND
DIRECTION


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A

B

If the magnitude of these two PARALLEL vectors is equal, are the two vectors the same?

Does ? What would we get if they were added ? A= B

DIRECTION MATTERS !!!

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II. ADDING VECTORS

•  Here’s the anatomy of a vector:

•  When vectors are added, you are finding what is called the RESULTANT vector. •  Vectors are always added by placing them head-to-tail

5 units, 0o

AADD:

3 units, 00

B

These vectors have the same direction. When I put them together, head to tail, the resultant vector will be 8 units long, 0o net direction (due East)

head-arrow end

tail-no arrow

direction = angle with x-axis (East), CCW length = magnitude

+ = 8 units, 00

A+ B= 8 units, 0

o

R

A. ADDING 1-DIMENSIONAL VECTORS

5 units, 0o

AADD:

3 units, 1800

B

These vectors have the opposite direction. When I put them together, head to tail, the resultant vector will be 2 units long, 0o net direction

Basically, because the vectors are in opposite directions, their magnitudes are opposite in sign. When added, the resultant has to be smaller.

If vector B had been larger than vector A, in which direction would the resultant point?

+ = 2 units, 00

A+ B= 2 units, 0

ovectors laid head to tail resultant is the sum

R

B. ADDING 2-DIMENSIONAL VECTORS

•  Let’s start with two vectors that form a 90o angle when added •  The vectors to be added are always placed head-to-tail •  The resultant vector is formed by connecting the tail of the first vector to the head of the last vector •  Pythagorean Theorem is used to compute the magnitude of the resultant •  The direction of the resultant is found using sin, cos, or tan relationships

8


9


a2 + b2 = 112 +112 = 242 = 15.6 km

Here we are adding two vectors with the same magnitude but directions that form a 90o angle when added.

head to tail 90o angle

MAGNITUDE DIRECTION

R

A

B

sinθ = opphyp

=1115.6

= 0.705

θ = sin−1 0.705 = 44.8o

θ

The direction is 45o which is what you’d expect from a triangle with the length of both sides being equal.

This is just an application of the Pythagorean theorem.

This is super easy to do because vector A only has a y-component and vector B only has an x-component.

Can you see, then, that one dimensional vectors are just the x or y component of their resultant?

sides = 11 units

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ADD: + =

A

B R

6 units, 0o 4 units, 270o A

B

MAGNITUDE

DIRECTION

vectors placed head to tail

R

B

A

The direction is the angle the resultant makes with an imaginary x-axis (East), measured counter clockwise.

Can you see the direction angle will be 270o + θ = 326.3o

a2 + b2 = 62 + 42 = 52 = 7.21 units

θ

tanθ = oppadj

=64= 1.5

θ = tan−11.5 = 56.3o

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Can you see that when two vectors are added that form a 90o angle, the horizontal vector is just the x-component of the resultant and the vertical vector is just the y-component of the resultant?

C. Adding two vectors that do not form a 90o angle when added

How would you add:

Place them head to tail, without disturbing their directions.

The order you place them makes no difference (adding vectors obeys the commutative and associative properties)

A= 5 units, 70o

R R

A

B

A

B

These two diagrams are the same

B= 3 units, 30o

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Both vectors A and B have x and y components.

The sum of the x-components of A and B equal the x-component of the resultant.

The sum of the y-components of A and B equal the y-component of the resultant.

A

Ax

Ay B

By Bx

R

Rx

Ry

sin 70o = opphyp

=Ay5

Ay = 5sin 70o = 4.70 units

cos70o = adjhyp

=Ax5

Ax = 5cos70o = 1.71 units

Rx = Ax + Bx = 4.30 units

Ry = Ay + By = 6.20 units

magnitude of the resultant

cos30o = adjhyp

=Bx3

Bx = 3cos30o = 2.59 units

R A

B

sin 30o = opphyp

=By3

By = 3sin 30o = 1.5 units

R = Rx2 + Ry

2 = 7.54 units

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DIRECTION OF THE RESULTANT

The RESULTANT vector has a magnitude of 7.54 units and a direction of 55.3o

Rx = Ax + Bx = 4.30 units

Ry = Ay + By = 6.20 units

R

Rx

Ry

R = Rx2 + Ry

2 = 7.54 units

tanθ = oppadj

=RyRx

=6.204.30

= 1.44

θ = tan−11.44 = 55.3o

MAGNITUDE OF THE RESULTANT

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D. Adding more than 2 vectors

•  This can be done graphically, which we will do in class. •  This can be done with math, just like we did in the last section

Let’s add:

A= 10 units,50o

B= 4 units,120o C

= 4.5 units, 310o

A

C B

R

Add them head to tail (order doesn’t matter). Connect the tail of the first vector with the head of the last one drawn. That’s the resultant.

Each of these vectors has an x and y component. We’re going to do what we did before…add up the x components and y components to get the resultant

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A

C B

R

cos50o = adjhyp

=Ax10

Ax = 10cos50o = 6.43 units

sin50o = opphyp

=Ay10

Ay = 5sin50o = 7.66 units

A

B

C

cos120o = adjhyp

=Ax4

Ax = 4 cos120o = −2 units

sin120o = opphyp

=Ay4

Ay = 4sin120o = 3.46 units

cos310o = adjhyp

=Ax4.5

Ax = 4.5cos310o = 2.89 units

For the resultant, just add up the x and y components and use Pythagoras (like we did for 2 vector addition)

sin 310o = opphyp

=Ay4.5

Ay = 4.5sin 310o = −3.45 units

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R

Rx = Ax + Bx + Cx = 7.32 units

Ry = Ay + By + Cx = 7.66 unitsR = Rx

2 + Ry2 = 10.59 units

tanθ = oppadj

=RyRx

=7.667.32

= 1.04

θ = tan−11.04 = 46.1o

MAGNITUDE

DIRECTION

A

C B

R

Let’s apply some common sense… Look at the vector drawing again. Does it make sense that vector B and C will essentially “cancel” each other? Go look at the values of the x and y components again on the previous page.

Does it makes sense, therefore, that the resultant and vector A should have similar magnitude and direction? Both the drawing and the math tell you the same thing !