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We will see in the next chapter that not only forces but also location vectors -- which are vectors pointing from one point in space to another -- are full-fledged vectors. And so are velocity and acceleration ( EMch 12 stuff).
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In the language of mathematics we address the forces
,
.... etc. as vectors and use the notation :
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If we have numerical values available for the components we would write :
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The arrow on top of the symbol
merely indicates that this
quantity is a vector. Many textbooks use bold/italics symbols
instead, like F1 because the arrows are costly to
print.
On the WWW I am in a similar situation and will use the arrow mode and bold/italics interchangeably.
Using this notation, Equation 3.4c (just above) is then written as
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The mathematician would read this as :
Add the vectorand Equation 3.4c states the rule as to how to do that :
Adding two vectors to each other results in a new vector the x-component of which is determined by the adding the x-components of the given vectors. (Same for the y- and z- component)
The physicist and we would say :
The forces
How about subtraction of vectors ?
Formally we would write :
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but how would we determine the components of
if the components of
and
are given ? Well, a requirement one could put forward here would be
the following :
If you subtract first from a vector
a vector
to obtain an intermediate vector
and than ADD
to that, you should come back
to
.
The only way how this can be accomplished
by using the following rule :
| Subtracting two vectors from each other results in a new vector the x-component of which is determined by subtracting the x-components of the given vectors. (Same for the y- and z- component) |
If the components of a vector
are given, its magnitude
(which is always non-negative) is calculated according to :
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If we wish to multiply a given vector, say
by a scalar, say "a", we obtain a new vector, say
.
We formally write :
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and the rule which goes along with that is the following :
| Multiplying a vector by a scalar results in a new vector the x-component of which is determined by multiplying the the x-component of the given vector by the scalar. (Same for the y- and z- component) |
At least for the case of the scalar being a positive integer this rule makes a lot of sense because it ties in nicely with the addition of vectors : whether you add a given vector 4 times to itself or whether you multiply the vector by 4 should give identical results :
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Speaking in terms of forces : 4 times
is a force
which is 4 times as strong as
and pointing in the same direction as
.
Similarily, in the following equation the left side and right side give identical results :
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Speaking in terms of forces : -1 times
is a force which has the same magnitude as
but is pointing exactly in opposite direction.
calculated according to Equation 4.3a then we can of course multiply
the vector
by the scalar 1/F where F=magnitude of
.
The result is a vector
which has a special property and therefore is given in many textbooks
a special symbol, the greek lower case lambda :
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Because the magnitude F (and with that 1/F) is never negative the vector
is pointing in the same direction as the vector
itself.
The unique property of
is that its magnitude is always 1 (one)
regardless of what values (and units) the components of
are (except if they are all zero). Because of this
property we call
a unit vector.
Among the infinitely many unit vectors which one can calculate there
are a few worth mentioning. If in Equation 4.5a the vector
has a positive x-component but zero y- and z-component, that is if it
is pointing along the positive x-axis, then
is pointing along the positive x-axis as well, still having the length one.
In this special case we often (and this is almost universal)
give that unit vector the symbol
. The analogs for the y- and z-directions
are given the symbols
and 
, respectively.
The unit vectors
,
, and
can be used to present any arbitrary vector
with components Fx, Fy, and Fz
in an alternative form :
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This way of writing out a vector will come in handy later on in the course, for now it is one example of how addition of vectors and multiplication with a scalar appear in a single equation.
and
is defined as :
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in which "F" and "r" denote the magnitude of
and
, respectively.
is the angle enclosed by the two vectors as shown in Figure 4.6a.
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If you know the two magnitudes and the angle you can use Equation 4.6a to determine the value (which may come out positive, negative, or zero) of the dot-product. |
If you know the components of the two vectors (let's say Fx, Fy,Fz, and rx,ry,rz) there is an alternative way to determine the value of the dot-product :
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Equations 4.6a and 4.6b can be used nicely to calculate unknown
quantities. For example if the components of the two vectors are
given and one wishes to know the angle between them, see
Prob. 4.9e.
In the 2-dimensional case it is not too difficult to derive Equation 4.6b
from 4.6a using only elementary geometry, see
Prob. 4.9j.
The vectors and
can represent any, even different
physical vector quantities.
From the point of view of physics
often represents a force acting on a body
while
represents the distance the same body moves. In this case the value of
the dot-product equals the amount of work the force performs on the body.
Here are two equations which come in handy at times :
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Another well-known case is when the two vectors
and
are one and the same. We find that for any vector
the dot-product with itself is equal to its magnitude squared :
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Lets assume you have given the components of a force
acting on a body of arbitrary shape
at some point B as shown in the figure to the left. The body is supported at point A by what we call a ball/socket joint which allows the body to swivel freely about point A. Lets call the vector from point A to point B the vector
and lets assume we know the value of its components.
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Question : About which axis will the given body start to rotate
under influence of the force
and how effective is the given force in trying to actually rotate the
body ?
In Chapter 5 we will do some investigating ourselves, for now I state :
and
.
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is the angle enclosed by the two vectors
and
.Well, if I gave you the components of these two vectors you could calculate the magnitudes F and r and use the dot-product to determine the angle between the two vectors and with that the value of M (= moment or torque in physics terms) is determined. The orientation of the axis A-C is harder to determine with the tools we have sofar at our disposal.
This is where the cross-product comes in.
| The cross-product of two vectors is a new vector which is perpendicular to the first two vectors and points in such a direction that it gives you also the direction of rotation. The length of this new vector is given by Equation 4.7a. |
Formally I write :
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but the problem is of course to determine the components of the
vector
for given
and
.
If you know what the determinant of a matrix is then the following says it all :
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or multiplying it all out :
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Examples of evaluating the cross-product of given vectors : Prob. 4.9f , and Prob. 4.9g , and Prob. 4.9h .
Klick here to do the test.
Prob. 4.9a :
Addition/subtraction of vectors
Prob. 4.9b : Multiplication by a scalar
Prob. 4.9c : Magnitude of Unit vector
Prob. 4.9d : Value of dot product
Prob. 4.9e :
Dot product -- Angle between vectors
Prob. 4.9f : Evaluate cross product
Prob. 4.9g : Proof on cross product
Prob. 4.9h :
Cross product -- Angle between vectors
Prob. 4.9i :
Broadcast pole, 3-D (Prob 3.6h revisited)
Prob. 4.9j :
Proof on dot-product
Zig Herzog, hgn@psu.edu
Last revised: 12/07/00
