6.Symposium on Explosives and Pyrotechnics, Organizer: Franklin Applied Physics, USA, every 3rd year, 1954 – 2000 (17th)
7.International Symposium on Loss Prevention and Safety Promotion in the Process Industries, Organizer: European Federation of Chemical Engineering, every 3rd year, 1974 – 2000 (10th)
8.International Pyrotechnics Seminar, Organizer: IPS (The International Pyrotechnics Society, USA), every year, 1968 – 2000 (27th)
9.International Symposium on Analysis and Detection of Explosives, Organizer: changing, every 3rd year, 1983 – 2001 (7th)
10.Explosives Safety Seminar, Organizer: Department of Defense Explosives Safety Board, every year, since 1974 every 2nd year, obtainable from National Technical Information Service (NTIS), US Department of Commerce), 1958 – 2001 (29th)
11.Airbag 2000 +: International Symposium on Sophisticated Car Occupant Safety Systems, Organizer: Fraunhofer-Institut Chemische Technologie (ICT), Germany, every 2nd year, 1992 – 2000 (5th)
12.Detection and Remediation Technologies for Mines and Minelike Targets, every year, Organizer: International Society for Optical Eng. – SPIE, 1996 – 2000 (5th)
13.Conference on Explosives and Blasting Technique, every year, Organizer: International Society of Explosives Engineers, 1975 – 2001 (27th)
14.International Autumn Seminar on Propellants, Explosives and Pyrotechnics: Theory and Practice of Energetic Materials, 1996 – 1999 (3rd)
15.Symposium on Explosives and Blasting Research, every year, Organizer: International Society of Explosives Engineers, 1985 – 1998 (14th)
The A to Z of Mathematics: A Basic Guide.
Thomas H. Sidebotham
Copyright ∂ 2002 John Wiley & Sons, Inc.
ISBN: 0-471-15045-2
V
VARIABLE
Reference: Constants.
VARIANCE
In statistics the variance of a set of data is the square of the standard deviation of the data:
Variance = σ 2
Reference: Standard Deviation.
VARIATION
There are two kinds of variation discussed in this text. One is direct variation, which is also called direct proportion, and the other is inverse variation, which is also called inverse proportion. Both of these terms are explained in the entry Proportion.
Reference: Proportion.
VECTOR
In this entry we discuss only two-dimensional vectors. Vectors are quantities that have both magnitude and direction. Examples of vectors are force, velocity, and acceleration. Speed is not a vector, because it has magnitude, but not direction. Suppose Ken is using a garden roller on the lawn and he is pulling with a force of 200 newtons. We have stated the magnitude of the force, but not the direction. We may add that Ken is pulling at an angle of 30◦ to the horizontal. Both the magnitude and the direction are needed to describe a vector quantity.
In this entry, a line segment represents a vector, and the length of the line segment represents the magnitude or size of the vector and the direction of the line segment is
VECTOR 465
the direction of the vector. Ken is pulling the garden roller with a force of 200 newtons at an angle of 30◦ with the lawn. This vector can be represented by a line segment which is drawn at an angle of 30◦ with the horizontal (see figure a). An arrow on the line indicates the direction in which the force acts. Figure a is a scale drawing of the vector where 200 newtons is represented by the length of the line segment.
200 N
30°
(a)
There are different ways of writing vectors. Suppose a vector is represented on the grid in figure b by a line segment AB. In writing by hand, it is difficult to express vectors by thick letters as when using heavy type, and this notation will not be used in this text. Writing the vector AB means that its direction is from A to B. The vector BA is the negative of AB, and its direction is from B to A.
B
A
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or AB or AB |
a or a |
F or F |
AB |
(b)
Vectors are also expressed as 2 by 1 matrices, in a similar way to translations. The vector AB can be expressed as AB = 43 , and the vector BA = −−43 . (For this positive and negative convention see the entry Translations.) When vectors are expressed as 2 by 1 matrices they are called column vectors, and can be added and subtracted in the following way.
Example 1. If a = −12 and b = −43 , work out (a) a + b, (b) a − b, and (c) 2a + 3b
Solution. For (a), write |
−3 |
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+ |
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= |
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1 |
+ |
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a |
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b |
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−2 |
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4 |
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= |
1−2 (+ |
4 |
Adding the numbers in the column vectors |
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3) |
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+ − |
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= −22 |
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Adding integers |
466 |
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VECTOR |
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For (b), write |
− −3 |
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− |
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= |
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1 |
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a |
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b |
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−2 |
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4 |
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= |
1−2 (− |
4 |
Subtracting the numbers in the column vectors |
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3) |
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− − |
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= |
4 |
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Subtracting integers |
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−6 |
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For (c), write
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+ |
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= |
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× |
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1 |
+ |
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× |
−3 |
2a |
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3b |
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2 |
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−2 |
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3 |
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4 |
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= |
2 |
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+ |
−9 |
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−4 |
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12 |
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= 8 −7
Multiplying the first vector by 2 and the second by 3
Adding the two vectors
The magnitude, or size, of a vector is the length of the line segment that represents the vector. It is found using the theorem of Pythagoras.
Example 2. Find the magnitude of the vector a = −43 .
−3 
a
4
(c)
Solution. The vector is drawn in figure c. Write
a2 = |
42 + (−3)2 |
Pythagoras’ Theorem |
a2 |
= |
16 + 9 |
Squaring the numbers |
a2 |
= |
25 |
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a = |
√ |
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25 |
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Taking square roots |
a = |
5 |
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The magnitude of the vector is 5.
Vectors can also be added using a vector triangle, as explained in the following example.
Example 3. Pat loves swimming and decides to swim across a stream that flows uniformly at a speed of 2 km/h. In still water Pat can swim at 3 km/h. Her plan is to swim directly across the stream at right angles to the bank, but the water pulls her downstream. What is her resultant speed, and in what direction does she cross the river?
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2 km/h |
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S |
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2 |
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S |
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θ |
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P |
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3 |
P |
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R |
R |
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RIVER |
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3 km/h
(d)
Solution. Let the velocity of the stream be S and the velocity of Pat be P. These two vectors are added using a vector triangle in the following way. Draw the vector P (see figure d). Draw the vector S starting at the point where P ends. Then complete the triangle of vectors with the resultant vector R as the hypotenuse. Draw an arrow on the vector R in the direction from where P starts to where S ends, as shown in the figure. Write
R2 = |
32 + 22 |
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Theorem of Pythgoras |
R2 = |
9 + 4 |
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R2 = |
13 |
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R = |
3.61 |
(to 2 dp) |
Taking the square root of 13 |
tan θ = |
23 |
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opposite |
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Tan = |
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, using trigonometry |
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adjacent |
θ = tan−1 |
2 |
If tan θ = a, then θ = tan−1a |
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3 |
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θ |
= |
56.3◦ |
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Using the calculator |
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Pat’s resultant speed is 3.61 km/h at an angle of 56.3◦ with the bank.
References: Components of a Vector, Line Segment, Pythagoras’ Theorem, Translation,
Trigonometry.
VECTOR TRIANGLE
References: Components of a Vector, Vector.
468 VERTICAL
VELOCITY
Velocity is defined to be the rate at which the displacement of an object is changing as time changes. The basic unit of velocity, as of speed, is meters per second. Another common unit is kilometers per hour. If an object is traveling at a constant speed of v meters/second for t seconds and covers d meters, then the formulas connecting these quantities are
distance = speed × time, |
speed = |
distance |
, |
time = |
distance |
time |
speed |
Example. Amanda runs the 100 meters in 12.9 seconds. Find her speed in meters/ second and in kilometers/hour, assuming she runs at a constant speed throughout.
Solution. Write
Speed = distance
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time |
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= |
100 |
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Substituting distance = 100, time = 12.9 |
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12.9 |
= 7.75 |
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(to 2 dp) |
Amanda’s speed is 7.75 meters/second.
Now write
Speed = 7.75 × 3.6 1 meter/second = 3.6 kilometers/hour
= 27.9
Amanda’s speed is 27.9 kilometers/hour.
Reference: Displacement.
VELOCITY–TIME GRAPHS
Reference: Acceleration.
VERTEX
Reference: Edge.
VERTICAL
VERTICALLY OPPOSITE ANGLES |
469 |
VERTICAL LINE TEST
Reference: Correspondence.
VERTICAL PLANE
Reference: Inclined Plane.
VERTICALLY OPPOSITE ANGLES
When two straight lines intersect at a vertex there are two pairs of congruent angles. Angles at the vertex that are opposite each other are called vertically opposite angles, and are equal in size. In this geometry theorem, “vertically” has no reference to the word vertical, but is derived from the word vertex (see figure a):
Angle a = angle b |
and |
Angle c = angle d |
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b c |
a |
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d |
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(a) |
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Example. Figure b shows an open pair of scissors. If angle x = 47◦, find the size of angle y.
x 
y
(b)
Solution. Write
y = 47◦ |
Vertically opposite angles are equal |
Reference: Geometry Theorems.
470 VOLUME
VOLUME
The volume of a solid shape is a measure of the three-dimensional space it occupies. It is measured in cubic units, which is written as units3. We can find the volume of a solid shape, say a cuboid, by counting the number of cubes that its three-dimensional space occupies. The volume of a cuboid measuring 3 cm by 2 cm by 3 cm can be found by counting the number of cubic centimeters (abbreviated cm3) it occupies. There are three layers of cubes, and in each layer there are six cubes, so the volume of the cuboid is 6 + 6 + 6 = 18 cm3.
height
width
length
Alternatively, the volume of the cuboid can be found using the formula (see the figure)
Volume = length × width × height |
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= 3 × 2 × 3 |
Substituting length = 3, width = 2, |
= 18 cm3 |
and height = 3 |
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For examples of finding the volumes of well-known solids see the respective entries. The units commonly used for volume are
Cubic millimeter, mm3
Cubic centimeter, cm3
Cubic meter, m3
The relationships between these units are
1000 mm3 = 1 cm3
1,000,000 cm3 = 1 m3
When we are finding the volume of liquid that a vessel holds we say we are finding the capacity of the vessel.
References: Capacity, Cube, Cuboid, Metric Units.
The A to Z of Mathematics: A Basic Guide.
Thomas H. Sidebotham
Copyright ∂ 2002 John Wiley & Sons, Inc.
ISBN: 0-471-15045-2
X
X-AXIS
Reference: Cartesian Coordinates.
X COORDINATE
Reference: Cartesian Coordinates.
The A to Z of Mathematics: A Basic Guide.
Thomas H. Sidebotham
Copyright ∂ 2002 John Wiley & Sons, Inc.
ISBN: 0-471-15045-2
Y
Y = MX + C
References: Cartesian Coordinates, Gradient-Intercept Form, Graphs.
YARD
Reference: Imperial System of Units.
