Concept Summary

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Topic | Discussion | Learning Aids
| 1.2 UNITS The SI system of units includes the meter (m), the kilogram (kg), and the second (s) as the base units for length, mass, and time, respectively. |
Meter | One meter is the distance that light travels in a vacuum in a time of 1/299 792 458 second. |
Kilogram | One kilogram is the mass of a standard cylinder of platinum–iridium alloy kept at the International Bureau of Weights and Measures. |
Second | One second is the time for a certain type of electromagnetic wave emitted by cesium-133 atoms to undergo 9 192 631 770 wave cycles. |
Conversion of units | 1.3 THE ROLE OF UNITS IN PROBLEM SOLVING To convert a number from one unit to another, multiply the number by the ratio of the two units. For instance, to convert 979 meters to feet, multiply 979 meters by the factor (3.281 foot/1 meter). |
Dimension | The dimension of a quantity represents its physical nature and the type of unit used to specify it. Three such dimensions are length [L], mass [M], time [T]. |
Dimensional analysis | A method for checking mathematical relations for the consistency of their dimensions. |
Sine, cosine, and tangent of an angle | 1.4 TRIGONOMETRY The sine, cosine, and tangent functions of an angle are defined in terms of a right triangle that contains :

where ho and ha are, respectively, the lengths of the sides opposite and adjacent to the angle , and h is the length of the hypotenuse.
Inverse trigonometric functions | The inverse sine, inverse cosine, and inverse tangent functions are

Pythagorean theorem | The Pythagorean theorem states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the other two sides:

| (1.7)

Scalars and vectors | 1.5 SCALARS AND VECTORS A scalar quantity is described completely by its size, which is also called its magnitude. A vector quantity has both a magnitude and a direction. Vectors are often represented by arrows, the length of the arrow being proportional to the magnitude of the vector and the direction of the arrow indicating the direction of the vector. |
Graphical method of vector addition and subtraction | 1.6 VECTOR ADDITION AND SUBTRACTION One procedure for adding vectors utilizes a graphical technique, in which the vectors to be added are arranged in a tail-to-head fashion. The resultant vector is drawn from the tail of the first vector to the head of the last vector. |
| The subtraction of a vector is treated as the addition of a vector that has been multiplied by a scalar factor of − 1. Multiplying a vector by −1 reverses the direction of the vector. |

Use Self-Assessment Test 1.1 to evaluate your understanding of Sections 1.5 and 1.6.

Vector components | 1.7 THE COMPONENTS OF A VECTOR In two dimensions, the vector components of a vector are two perpendicular vectors and that are parallel to the x and y axes, respectively, and that add together vectorially so that . |
Scalar components | The scalar component Ax has a magnitude that is equal to that of and is given a positive sign if points along the axis and a negative sign if points along the −x axis. The scalar component Ay is defined in a similar manner. |
Condition for a vector to be zero | A vector is zero if, and only if, each of its vector components is zero. |
Condition for two vectors to be equal | Two vectors are equal if, and only if, they have the same magnitude and direction. Alternatively, two vectors are equal in two dimensions if the x vector components of each are equal and the y vector components of each are equal. |
| 1.8 ADDITION OF VECTORS BY MEANS OF COMPONENTS If two vectors and are added to give a resultant such that , then

where Cx,Ax, and Bx are the scalar components of the vectors along the x direction, and Cy, Ay, and By are the scalar components of the vectors along the y direction.

Use Self-Assessment Test 1.2 to evaluate your understanding of Sections 1.7 and 1.8.

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