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Weather Observation and Analysis
John NielsenGammon
Course Notes
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Chapter 7. BITS OF VECTOR CALCULUS
7.1 Vector Magnitude and Direction
Consider the vector shown in the diagram.
The vector is drawn
pointing toward the upper right.
The origin of the vector is, literally, the
origin on this
xy
plot.
Suppose we want to know the magnitude of this vector.
In high
school you probably learned about computing vector lengths by starting
ATMO 251
Chapter 7: Vector Calculus
page 1 of 21
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squares of the lengths, summing them, and then taking the square root.
Well, we’ll use that technique eventually, but that’s way too complicated
for most weather analysis applications.
Instead, let’s keep things simple.
The length of the vector is
proportional to its magnitude, so once we know what a given vector length
corresponds to, we can just measure the vector and convert it to a
magnitude.
Since the figure in this case has a grid background, we’ll start
by asking what the magnitude of a vector would be if it were exactly one
grid box side long.
Then we can see how many grid boxes the vector
covers, and convert that to a vector magnitude.
Enough hypotheticals, let’s do this for real.
Let’s say the vector is
the horizontal wind.
The magnitude of the wind is called the wind speed.
Now suppose each grid box corresponds to a wind speed of one meter per
second (1 m s
1
).
If we take a ruler to the page, we find that each grid box
is half an inch wide.
So a vector that’s ½ inch long on this particular
graph would have a magnitude of 1 m s
1
.
A vector that’s an inch long
would be 2 m s
1
, a vector that’s 1 ½ inches long would be 3 m s
1
, and so
forth.
If we measure the vector, we find that the vector is 1.6 inches long.
So the wind speed is just a little bit more than 3 m s
1
.
Specifically, it’s
3.2 m s
1
.
For some of you, it may be obvious where that answer came from.
If not, this is like any conversion problem, and solving conversion
ATMO 251
Chapter 7: Vector Calculus
page 2 of 21
problems is a necessary skill, so let’s work it out in detail.
There is one
conversion here:
1 m s
1
= 0.5 graph inches
In this case, we know the length of the vector in graph inches.
Divide the
conversion equation by the side with the units that have been measured:
1 m s
1
/ 0.5 graph inches = 0.5 graph inches / 0.5 graph inches
The right hand side is unity: a number divided by itself.
Divide and
simplify the left hand side so that its denominator is unity too:
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This note was uploaded on 04/02/2008 for the course ATMO 251501/50 taught by Professor Alcorn during the Fall '07 term at Texas A&M.
 Fall '07
 Alcorn

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