Unit 6 text 6.1

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Essential Vocabulary

cone – конус intersect – пересікати(ся)

cube – куб pyramid – піраміда

cuboid – прямокутний паралелепіпед rectangular – прямокутний

cylinder – циліндр triangular – трикутний

edge – грань; ребро vertex – вершина

Anything, which takes up space, is spoken of as a solid. Thus each page of this book is a solid, however thin the paper may be. The word solid as used here must not be confused with the word solid, which is used as opposed to liquid and gas.

Most solids are irregular in shape, e.g. a pebble in a stream, a cloud in the sky. Geometry deals with the shape, size, and position of solids, which are regular in shape, e.g. a ball, a matchbox, a pencil.

The more common regular solids are: cube, cuboid or rectangular prism, triangular prism, square pyramid, cylinder, cone, sphere.

Solids are bounded by surfaces. These surfaces separate the solids from the surrounding space. Surfaces are of two kinds: plane and curved. The surfaces of a cube, rectangular prism, and pyramid are plane surfaces, while the surface of a sphere is curved. A sheet of paper, e.g. a leaf of a book, may represent a surface, but even the thinnest sheet of paper will be a geometrical solid, since it has length, breadth, and thickness.

Surface intersected in lines are bounded by lines. Lines are either straight or curved.

Lines intersect in points. The meeting place of two edges is called a point (a vertex). The dot made on paper by a fine pencil point represents a point. No matter how fine the pencil point is, however, the dot is a geometrical solid since it has length, breadth, and thickness, and a point has no length, no breadth, and no thickness. A point indicates but has no size.

Exercise 6.1

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The genesis of the theory of quadratic forms lies in analytic geometry, namely, in the theory of quadratic curves and surfaces. It will be recalled that the equation of a central quadratic curve in a plane, after translating the origin of the rectangular coordinate system to the centre of the curve, is of the form

Ax² + 2Bxy + Cy² = D (1)

It is also possible to perform a rotation of the coordinate axes through an angle α, such that we have the following transformation from the coordinates x, y to the coordinates x´, y´:

x = x´ cos α – y´ sin α

y = x´ sin α + y´ cos α

Then the equation of our curve in the new coordinates will be of “canonical” form:

A´x´² + C´y´² = D (2)

In this equation, the coefficient of the product of unknowns x´y´ is, thus, zero. The transformation of coordinates may obviously be interpreted as a linear transformation of the unknowns; the transformation is non-singular since the determinant of its coefficients is equal to unity.

Exercise 6.2

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Направлений відрізок (або упорядкована пара точок) називається вектором (геометричним). До векторів також відноситься так званий нульовий вектор, у якого початок і кінець співпадають. Відстань між початком і кінцем вектора називається його довжиною або модулем і позначається |α|. Модуль нульового вектора дорівнює нулю.

Вектори, що знаходяться на одній прямій або на паралельних прямих, називаються колінеарними.

Вектори називаються компланарними, якщо існує площина, якій вони паралельні.

Два вектори будемо вважати рівними, якщо вони колінеарні, однаково направлені та мають рівні довжини. Проекція вектора α на вісь L визначається як pr_L α = |α | cos φ, де φ – кут між вектором і віссю L.

TEXT 6.2

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Essential Vocabulary

straight line – пряма лінія surface – поверхня