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References: Binomial, Degree, Quadratic Equations.
References: Concave, Maximum Value, Stationary Point.
Reference: Metric Units.
References: Cumulative Frequency, Interquartile Range.
In statistics the variance of a set of data is the square of the standard deviation of the data:
Variance = a2
Reference: Standard Deviation.
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.
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
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.
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
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
a — b
For (b), write
-2-4 1 - (-3)
Subtracting the numbers in the column vectors Subtracting integers
For (c), write
2a + 3b = 2x[ ) + 3 x
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
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.
R2 = 32 + 22 R2 = 9 + 4 R2 = 13
Theorem of Pythgoras
R = 3.61 (to 2 dp) Taking the square root of 13
tan# = §
If tan 6 = a, then 0 = tan 1a
0 = 56.3°
Using the calculator
References: Components of a Vector, Vector.
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