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Karnataka Class 12 Commerce Maths Vectors Complete Information

Karnataka Class 12 Commerce Maths Vectors Complete Information

Karnataka Class 12 Commerce Maths Vectors : 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.

Karnataka Class 12 Commerce Maths Vectors Complete Information

Karnataka Class 12 Commerce Maths Vectors : Here we provides you Karnataka Class 12 Commerce Maths Integrals Complete Notes in PDF Format. Karnataka Class 12 Commerce Maths Integrals topic are  Vectors and scalars, magnitude and direction of a vector.Direction cosines and direction ratios of a vector. Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio. Definition, Geometrical Interpretation, properties and application of scalar (dot) product of vectors, vector (cross) product of vectors, scalar triple product of vectors.

Download here Karnataka Class 12 Commerce Maths Complete Notes In PDF Format 

Karnataka Class 12 Commerce Maths Vectors Complete Information

Karnataka Class 12 Commerce Maths Vectors : Examples of such quantities include distance, displacement, speed, velocity, acceleration, force, mass, momentum, energy, work, power, etc. All these quantities can by divided into two categories – vectors and scalars. A vector quantity is a quantity that is fully described by both magnitude and direction. From his unusual beginning in “Defining a vector” to his final comments on “What then is a vector?” author Banesh Hoffmann has written a book that is provocative and unconventional. In his emphasis on the unresolved issue of defining a vector, Hoffmann mixes pure and applied mathematics without using calculus. The result is a treatment that can serve as a supplement and corrective to textbooks, as well as collateral reading in all courses that deal with vectors.

Major topics include vectors and the parallelogram law; algebraic notation and basic ideas; vector algebra; scalars and scalar products; vector products and quotients of vectors; and tensors. The author writes with a fresh, challenging style, making all complex concepts readily understandable. Nearly 400 exercises appear throughout the text. Professor of Mathematics at Queens College at the City University of New York, Banesh Hoffmann is also the author of The Strange Story of the Quantum and other important books. This volume provides much that is new for both students and their instructors, and it will certainly generate debate and discussion in the classroom.

Karnataka Class 12 Commerce Maths Vectors Complete Information

Karnataka Class 12 Commerce Maths Vectors :  A vector is a mathematical object that has magnitude and direction. With other words it is a line of given length and pointing along a given direction. The magnitude of vector vector a is its length and is denoted by |vector a|.

If two vectors vector a,vector b are in the same direction then vector a = n.vector b where n is a real number.

if 0 < n < 1 then |vector a| < |vector b|
if 1 < n then |vector a| > |vector b|
if n < 0 then vector a || vector b and the direction of vector a is opposite the direction of vector b

Addition of two vectors is accomplished by laying the vectors head to tail in sequence to create a triangle such as is shown in the figure.

A vector can be resolved along any two directions in a plane containing it. The figure shows how the parallelogram rule is used to construct vectors vector a and vector b that add up to vector c.

Vector scalar product

Let’s have two vectors. Vector scalar product is the formula:

other notations for scalar product is vector avector b or (vector a,vector b)
The result from scalar product of two vectors is always a real number.

Scalar product properties

  • vector avector b = vector bvector a
  • n(vector avector b) = (nvector a)vector b = vector a(nvector b) where n is number
  • vector a(vector b + vector c) = vector avector b + vector avector c

If the angle between two verctors vector a,vector b is 90° then vector avector b = 0, because cos(90°) = 0
vector avector a = |vector a|2 because the angle between 2 vectors vector a is 180° and cos(180°) = 1

Vectors Problems

1) If vector a = -1.vector b what can we say about those two vectors?
Solution: Those two vectors are parallel, with the same magnitude and point to contrary directions.

2) What is the scalar product vector avector b if |vector a| = 5, |vector b| = 7 and the angle between the two vectors is 30°

3) Prove with vectors that for every triangle the lenght of one side is smaller than the sum of the other two sides.

Karnataka Class 12 Commerce Maths Vectors Complete Information

Karnataka Class 12 Commerce Maths Vectors : This statement may provide yourself enough information to pique your interest; yet, there is not enough information included in the statement to find the bag of gold. The displacement required to find the bag of gold has not been fully described. On the other hand, suppose your teacher tells you “A bag of gold is located outside the classroom. To find it, displace yourself from the center of the classroom door 20 meters in a direction 30 degrees to the west of north.” This statement now provides a complete description of the displacement vector – it lists both magnitude (20 meters) and direction (30 degrees to the west of north) relative to a reference or starting position (the center of the classroom door). Vector quantities are not fully described unless both magnitude and direction are listed.

Representing Vectors

Vector quantities are often represented by scaled vector diagrams. Vector diagrams depict a vector by use of an arrow drawn to scale in a specific direction. Vector diagrams were introduced and used in earlier units to depict the forces acting upon an object. Such diagrams are commonly called as free-body diagrams. An example of a scaled vector diagram is shown in the diagram at the right. The vector diagram depicts a displacement vector. Observe that there are several characteristics of this diagram that make it an appropriately drawn vector diagram.

  • a scale is clearly listed
  • a vector arrow (with arrowhead) is drawn in a specified direction. The vector arrow has a head and a tail.
  • the magnitude and direction of the vector is clearly labeled. In this case, the diagram shows the magnitude is 20 m and the direction is (30 degrees West of North).

 Conventions for Describing Directions of Vectors

Vectors can be directed due East, due West, due South, and due North. But some vectors are directed northeast (at a 45 degree angle); and some vectors are even directed northeast, yet more north than east. Thus, there is a clear need for some form of a convention for identifying the direction of a vector that is not due East, due West, due South, or due North. There are a variety of conventions for describing the direction of any vector. The two conventions that will be discussed and used in this unit are described below:

1) The direction of a vector is often expressed as an angle of rotation of the vector about its “tail” from east, west, north, or south. For example, a vector can be said to have a direction of 40 degrees North of West (meaning a vector pointing West has been rotated 40 degrees towards the northerly direction) of 65 degrees East of South (meaning a vector pointing South has been rotated 65 degrees towards the easterly direction).

2) The direction of a vector is often expressed as a counterclockwise angle of rotation of the vector about its “tail” from due East. Using this convention, a vector with a direction of 30 degrees is a vector that has been rotated 30 degrees in a counterclockwise direction relative to due east. A vector with a direction of 160 degrees is a vector that has been rotated 160 degrees in a counterclockwise direction relative to due east. A vector with a direction of 270 degrees is a vector that has been rotated 270 degrees in a counterclockwise direction relative to due east. This is one of the most common conventions for the direction of a vector and will be utilized throughout this unit.

Two illustrations of the second convention (discussed above) for identifying the direction of a vector are shown below.

Observe in the first example that the vector is said to have a direction of 40 degrees. You can think of this direction as follows: suppose a vector pointing East had its tail pinned down and then the vector was rotated an angle of 40 degrees in the counterclockwise direction. Observe in the second example that the vector is said to have a direction of 240 degrees. This means that the tail of the vector was pinned down and the vector was rotated an angle of 240 degrees in the counterclockwise direction beginning from due east. A rotation of 240 degrees is equivalent to rotating the vector through two quadrants (180 degrees) and then an additional 60 degrees into the third quadrant.

Representing the Magnitude of a Vector

The magnitude of a vector in a scaled vector diagram is depicted by the length of the arrow. The arrow is drawn a precise length in accordance with a chosen scale. For example, the diagram at the right shows a vector with a magnitude of 20 miles. Since the scale used for constructing the diagram is 1 cm = 5 miles, the vector arrow is drawn with a length of 4 cm. That is, 4 cm x (5 miles/1 cm) = 20 miles.

Using the same scale (1 cm = 5 miles), a displacement vector that is 15 miles will be represented by a vector arrow that is 3 cm in length. Similarly, a 25-mile displacement vector is represented by a 5-cm long vector arrow. And finally, an 18-mile displacement vector is represented by a 3.6-cm long arrow. See the examples shown below.

In conclusion, vectors can be represented by use of a scaled vector diagram. On such a diagram, a vector arrow is drawn to represent the vector. The arrow has an obvious tail and arrowhead. The magnitude of a vector is represented by the length of the arrow. A scale is indicated (such as, 1 cm = 5 miles) and the arrow is drawn the proper length according to the chosen scale. The arrow points in the precise direction. Directions are described by the use of some convention. The most common convention is that the direction of a vector is the counterclockwise angle of rotation which that vector makes with respect to due East.

Download here Karnataka Class 12 Commerce Maths Vectors Complete Information In PDF Format 

Karnataka Class 12 Commerce Maths Vectors Complete Information

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