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CBSE Class 12 Commerce Mathematics Three Dimensional Geometry Complete Notes

CBSE Class 12 Commerce Mathematics Three Dimensional Geometry Complete Notes

CBSE Class 12 Commerce Mathematics Three Dimensional Geometry  : CBSE is a renowned educational Board, which comes under the Union Government of India. This eminent board was formed in 1952 and associated with the Board of High School and Intermediate Education, Rajputana. Ajmer, Gwalior, Merwara and Central India were included in the administrative territory of this board along with the other places including Bhopal, Ajmer and Vindhya Pradesh. From 1952 onwards, it has been providing a standard education and robust learning environment to all. The Central Board of Secondary Education or CBSE is a prestigious board of education and it provides affiliation to public and private schools. Apart from this, all Jawahar Navodaya Vidyalayas and kendriya vidyalayas are affiliated to this board.

To impart quality education to its learners, CBSE took the required steps and also provides a healthy and holistic school education, which gives students adequate space to develop physically and mentally. The board conducts research and based on that, it evaluates its syllabus and educational pattern. CBSE provides standard education to all and also promotes a state-of-the art environment that makes students vivacious and competent in all aspects. CBSE Syllabus is well-structured as several proficient subject experts are associated with this board. The syllabi of CBSE Maths, CBSE Science along with other syllabi are amended from time to time to make students up-to-date with current information so that they can meet all educational demands confidently.

CBSE Class 12 Commerce Mathematics Three Dimensional Geometry Complete Notes

CBSE Class 12 Commerce Mathematics Three Dimensional Geometry  :  Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e., point). This is the informal meaning of the term dimension.

In physics and mathematics, a sequence of n numbers can be understood as a location in n-dimensional space. When n = 3, the set of all such locations is called three-dimensional Euclidean space. It is commonly represented by the symbol 3. This serves as a three-parameter model of the physical universe (that is, the spatial part, without considering time) in which all known matter exists. However, this space is only one example of a large variety of spaces in three dimensions called 3-manifolds. In this classical example, when the three values refer to measurements in different directions (coordinates), any three directions can be chosen, provided that vectors in these directions do not all lie in the same 2-space (plane). Furthermore, in this case, these three values can be labeled by any combination of three chosen from the terms width, height, depth, and breadth.

CBSE Class 12 Commerce Mathematics Three Dimensional Geometry Complete Notes

CBSE Class 12 Commerce Mathematics Three Dimensional Geometry  :  CBSE Class 12 Commerce  mathematics, analytic geometry (also called Cartesian geometry) describes every point in three-dimensional space by means of three coordinates. Three coordinate axes are given, each perpendicular to the other two at the origin, the point at which they cross. They are usually labeled x, y, and z. Relative to these axes, the position of any point in three-dimensional space is given by an ordered triple of real numbers, each number giving the distance of that point from the origin measured along the given axis, which is equal to the distance of that point from the plane determined by the other two axes.

Other popular methods of describing the location of a point in three-dimensional space include cylindrical coordinates and spherical coordinates, though there are an infinite number of possible methods. See Euclidean space.

Download here CBSE Class 12 Commerce Mathematics Three Dimensional Geometry Complete Notes in pdf format

CBSE Class 12 Commerce Mathematics Three Dimensional Geometry Complete Notes

CBSE Class 12 Commerce Mathematics Three Dimensional Geometry  : An object with three dimensions (such as height, width and depth) like any object in the real world.


Example: your body is three-dimensional.

Also known as “3D”

In three dimensions, the coordinate axes of a rectangular Cartesian co-ordinate system are three mutually perpendicular lines. The axes are called the x, y and z axes. The three planes determined by the pair of axes are the coordinate planes are called XY, YZ and ZX planes. The three coordinate planes divide the space into eight parts known as octants.

Co-ordinates of a Point

The coordinates of a point P in three dimensional geometry is always written in the form of triplet like (x, y, z). Here x, y and z are the distances from the YZ, ZX and XY planes.

  • Any point on x-axis is of the form (x, 0, 0)
  • Any point on y-axis is of the form (0, y, 0)
  • Any point on z-axis is of the form (0, 0, z)

Distance Between Two Points

Distance between two points (x1, y1, z1) and (x2, y2  z2) is given by

Direction Cosines

Direction cosines of a line are the cosines of the angles made by the line with the positive directions of the coordinate axes. If l, m, n are the direction cosines of a line, then

l2 + m2 + n2 = 1

Direction cosines of a line joining two points P(x1, y1, z1) and Q(x2, y2, z2) are

Direction Ratios

Direction ratios of a line are the numbers which are proportional to the direction cosines of a line. If l, m, n are the direction cosines and a, b, c are the direction ratios of a line, then

Skew Lines

Skew lines are lines in space which are neither parallel nor intersecting. They lie in different planes.

Angle Between Skew Lines

Angle between skew lines is the angle between two intersecting lines drawn from any point (or origin) parallel to each of the skew lines.

CBSE Class 12 Commerce Mathematics Three Dimensional Geometry Complete Notes

CBSE Class 12 Commerce Mathematics Three Dimensional Geometry  : Significant advances in three-dimensional electrocardiography have made this modality a powerful diagnostic tool in the cardiology clinic. It can provide accurate and reliable measurements of chamber size and function, including the quantification of left ventricular mechanical synchronous to guide patient selection for cardiac synchronize-satiation therapy. Furthermore, three-dimensional electrocardiography offers novel views and comprehensive anatomic definition of valvular and congenital abnormalities, improving diagnosis and preoperative planning. In addition, it is extremely useful in monitoring the effectiveness of surgical or per cutaneous trans catheter interventions. As its efficacy for more and more clinical applications is demonstrated, it is clear that three-dimensional electrocardiograph has become part of the routine clinical diagnostic armament. In this article, we describe the development of three-dimensional electrocardiograph over the last decades, review the scientific evidence for its current clinical use and discuss potential future applications. (Neth Heart J 2009;17:18-24.).

Ability in the clinic, emergency ward and operating room is a major advantage. Furthermore, its un com- placated use in children, pregnant women, and those with implanted pacemakers or defibrillators has set it apart from other closely competing imaging modalities, such as cardiovascular computed tomography (CT) and magnetic resonance imaging (MRI). However, both M mode and two dimensional electrocardiography (2DE) have their own limitations, mainly that quantitative measurements such as left ventricular (LV) volumes and ejection fraction (EF) are derived from assumptions of symmetrical LV geometry that are unreliable, especially in patients with ventricles that are dilated, aneurysm or with regional wall motion abnormalities. These assumptions as well as the inadvertent use of foreshortened views, and the conscious decision of many cardiology departments to rely on visual estimates of LV function and EF, despite the inherent subjectivity in interpreting two- dimensional (2D) data, probably account for the relative inaccuracy and poor reproducible of these techniques.

As important diagnostic, prognostic and therapeutic decisions rest upon this analysis, it is paramount that inter observer and nonobservant variability are reduced by developing and utilizing accurate and reproducible electrocardiograph quaint- fiction methods. A significant development in this regard has been the introduction of three-dimensional echo car- diography (3DE). Although the technology had a slow initial development with poor image quality and difficult and time-consuming acquisition and offline reconstruction techniques, through collaboration between clinical investigators and industry, great advancements in image acquisition and computer processing technology have been made to overcome these limitations. With the current state of technology, namely real-time 3D electrocardiography, and the high ongoing pace of development, 3DE will become available for an expanding range of clinical applications. Here, we review the development of 3DE from past to present, provide the scientific evidence for its clinical use and discuss its potential future appli- cations.

Keywords: cardiovascular imaging, real-time three-dimensional electrocardiography, clinical application,history, development Electrocardiography is the most clinically used diag-nostic imaging modality in cardiac practice today,and with good reason. In addition to its non invasive nature and time- and cost-efficiency, its bedside avail Clinical application of three-dimensional electrocardiography.

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