# The A to Z of mathematics a basic guide - Sidebotham T.H.

ISBN 0-471-15045-2

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References: Binomial, Degree, Quadratic Equations.

TRIOMINO

Reference: Polyominoes.

TURNING POINTS

References: Concave, Maximum Value, Stationary Point.

u

UNITS

Reference: Metric Units.

UNITY

One.

UPPER QUARTILE

References: Cumulative Frequency, Interquartile Range.

463

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 = a2

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

464

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.

(a)

There are different ways of writing vectors. Suppose a vector is represented on the grid in figure b by a line segment AS. 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 AS means that its direction is from A to B. The vector BA is the negative of AS, and its direction is from S to A.

AB or AB or AB

a or a

(b)

F or F

Vectors are also expressed as 2 by 1 matrices, in a similar way to translations. The vector AS can be expressed as AS = (3), and the vector SA = (~3). (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 = (~2) and& = (_^3), work out (a) a + b, (b) a—b, and (c) 2a + 3b Solution. For (a), write

Adding the numbers in the column vectors Adding integers

466 VECTOR

a — b

For (b), write

+G

-2-4 1 - (-3)

Subtracting the numbers in the column vectors Subtracting integers

For (c), write

2a + 3b = 2x[ ) + 3 x

-4\ /12

+

-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 = ( *3).

Solution. The vector is drawn in figure c. Write

a2 — 42 + (—3)2 Pythagoras’ Theorem

a2 = 16 + 9 Squaring the numbers

a2 =25

a = V25 Taking square roots

a = 5

The magnitude of the vector is 5.

Vectors can also be added using a vector triangle, as explained in the following example.

VECTOR TRIANGLE 467

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?

Solution. Let the velocity of the stream be 5 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 5 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 5 ends, as shown in the figure. Write

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.

2 km/h

S

2

S

P

RIVER

3 km/h

JZL

R2 = 32 + 22 R2 = 9 + 4 R2 = 13

Theorem of Pythgoras

R = 3.61 (to 2 dp) Taking the square root of 13

tan# = §

using trigonometry

If tan 6 = a, then 0 = tan 1a

0 = 56.3°

Using the calculator

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

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