## Karnataka Class 12 Commerce Maths Unit IV – Victors And Three Dimensional Geometry Notes

Karnataka Class 12 Commerce Maths Unit IV – Victors And Three Dimensional Geometry : CBSE board has been offering a robust, holistic school education to students since its inception. The board first analyzes students’ learning requirements and according to that, it prepares suitable syllabus for each class. Additionally, the board designs appropriate question papers to evaluate students’ subject knowledge at the end of each academic session. Moreover, to keep students stress free during exams, the board also designs sample papers for each class. The CBSE board conducts research to get to know the current educational requirements and based on that, it chooses suitable subjects and their relevant topics. Hence, students, who are pursuing their studies under this board, get updated information and keep them prepared for any competitive exams.

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

### Karnataka Class 12 Commerce Maths Unit IV – Victors And Three Dimensional Geometry Notes

Karnataka Class 12 Commerce Maths Unit IV – Victors And Three Dimensional Geometry : Here our team members provides you Karnataka Class 12 Commerce Maths Unit IV – Victors And Three Dimensional Geometry Notes in pdf format. Here we gave direct links for you easy to download Karnataka Class 12 Commerce Maths Unit IV – Victors And Three Dimensional Geometry Notes. Download these Karnataka Class 12 Commerce Maths Unit IV – Victors And Three Dimensional Geometry Notes and read well. Karnataka Class 12 Commerce Maths Unit IV – Victors And Three Dimensional Geometry Notes topics are :

**1. Vectors**

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.

**2. Three – dimensional Geometry**

Direction cosines and direction ratios of a line joining two points.Cartesian equation and vector equation of a line, coplanar and skew lines, shortest distance between two lines.Cartesian and vector equation of a plane.Angle between (i) two lines, (ii) two planes, (iii) a line and a plane.Distance of a point from a plane.

### Karnataka Class 12 Commerce Maths Unit IV – Victors And Three Dimensional Geometry Notes

Karnataka Class 12 Commerce Maths Unit IV – Victors And Three Dimensional Geometry : Karnataka Class 12 Commerce Maths Unit IV – Victors And Three Dimensional Geometry have two main Topics those explained Detail in bellow :

#### 1) Karnataka Class 12 Commerce Maths Vectors Complete Information

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.

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 is its length and is denoted by ||.

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

if 0 < n < 1 then || < ||

if 1 < n then || > ||

if n < 0 then || and the direction of is opposite the direction of

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 and that add up to .

#### Vector scalar product

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

other notations for scalar product is or (,)

**The result from scalar product of two vectors is always a real number.**

#### Scalar product properties

- =
- n() = (n) = (n) where n is number
- ( + ) = +

If the angle between two verctors , is 90° then = 0, because cos(90°) = 0

= ||^{2} because the angle between 2 vectors is 180° and cos(180°) = 1

#### Vectors Problems

1) If = -1. 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 if || = 5, || = 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.

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

#### 2) Karnataka Class 12 Commerce Maths Three-Dimensional Geometry Complete Notes

In 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.

Karnataka Class 12 Commerce Maths Three-Dimensional Geometry : In this chapter we present a vector–algebra approach to three–dimensional geometry. The aim is to present standard properties of lines and planes, with minimum use of complicated three–dimensional diagrams such as those involving similar triangles. We summarize the chapter: Points are defined as ordered triples of real numbers and the distance between points P1 = (x1, y1, z1) and P2 = (x2, y2, z2) .

By taking a look at parallelism and perpendicularity in three dimensions, we see that many of the same principles apply but that several complexities are also added.

**Key Terms**

o Skew lines

o Axes

o Cross section

**Objectives**

o Recognize perpendicular, parallel, and skew lines in three dimensions

o Be able to represent simple three-dimensional figures using two-dimensional drawings

o Know what a cross section is and how it can be used to relate two- and three-dimensional geometry

Objects in real-life have three spatial dimensions, so it is helpful for us to understand some of the fundamental aspects of three-dimensional geometry. In many ways, three-dimensional geometry is just an extension of two-dimensional geometry; many of the same principles apply. In this article, we consider parallelism and perpendicularity in three dimensions, and we then take a look at drawing three-dimensional figures on a two-dimensional surface. Finally, we look briefly at cross sections.

*Parallelism and Perpendicularity in Three Dimensions*

In two dimensions, we noted that any two lines that do not intersect are parallel. Recall, however, that we gave a slightly more fundamental definition that involved the relationship of the angles formed by a transversal that cuts across the parallel lines. Now, imagine two lines in three dimensions. In three dimensions, you can probably easily imagine two lines that are not really parallel but that do not intersect, either. An example is shown below using lines *l* and *m*, where one of the lines (*l*) is shown “on end” (that is, going into and out of the surface of the page) so that it appears as a point.

Notice that the two lines do not have any point of intersection, and yet, they are not what we would consider “parallel.” These are what we call **skew lines. **Note also that these lines, although they look in some sense like they are “perpendicular,” are not truly so because they do not intersect. If they did intersect, however, then they would indeed be perpendicular, as shown below.

In two-dimensional (planar) geometry, for a given point on a line, there exists only one perpendicular line, as shown below.

Karnataka Class 12 Commerce Maths Three-Dimensional Geometry : In three-dimensional geometry, there exist an infinite number of lines perpendicular to a given line. Consider a line *l* that intersects a plane at a right angle (in other words, wherever an angle measurement is taken around the line with respect to the plane, it is always 90°). We can draw innumerable lines in the plane that intersect line *l*; because they lie in the plane, they intersect *l* at a right angle.

Likewise, in planar geometry, there exist only two lines that are parallel to a given line for some fixed distance *d*, as shown below.

Karnataka Class 12 Commerce Maths Three-Dimensional Geometry : In three-dimensional geometry, once again, there exist an infinite number of lines parallel to a given line for some fixed distance *d*. Imagine the line *l* passing at a right angle through a plane once again, as shown above. This time, however, let’s draw a circle with radius *d* in the plane.

Now, we can draw any number of lines that intersect the circle and the plane at a right angle-these lines are also parallel to *l*.

### Download here Karnataka Class 12 Commerce Maths Three-Dimensional Geometry Complete Notes in PDF Format

### Karnataka Class 12 Commerce Maths Unit IV – Victors And Three Dimensional Geometry Notes

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