Vector Analysis 1.1 Elementary Approach

In most sciences one generation tears down what another has built and what one has established another undoes. In Mathematics alone each generation builds a new story to the old structure. – HERMAN HANKEL

thetransformationequations of coordinates the transformation equations of vectors Euclidean Geometry ans Newtonian physics Philosophical commenrts, Aristotle

For almost problems in classical dynamics, we can use the Newtonian mechanics in 3-dimensions to solve it but newtonian mechanics has it own limitations .As a matter of fact, Newton's first law is related to a frame of reference. Because generally, the acceleration of a body measured with reference to a frame of reference depends on that reference. The first law states that, if there is nobody available nearby, then a family of frame of reference could be obtained whereat no acceleration of a particle exists. Newton's first law says that if there is acceleration, there is a force, but there are no forces acting on the object because we know it is static. Major

Mathematics knows a several ways the multiplication of two, three or more vectors , all explained here. The two multiplying of the first group are the scalar and vector product – named according to the results, or the dot and a cross product of the vector – called according to the operation signs. We will consider them mainly geometrically, together with the 'communicator multiplication' – my private from the same group. Other multiplications have more vectors and are defined by the preceding ones, aggregating the factors as additional vectors. The text is an easier excerpt of the book of Quantum Mechanics, which this article is a promotion. 1 Scalar product The scalar product of two vectors, two oriented segments, a and b is a scalar a ⋅ b = ab cos γ, (1) where a = a and b = b are the intensities (lengths) of these vectors, and γ = ∠(a, b) is the angle between them. In other words, the scalar product is the product of the length of the first vector and the length of the projection of the second vector at first, as seen in the figure 1. In the wider theory of vectors, the scalar product is often called internal product, and dot product too.

Physical Review Special Topics - Physics Education Research, 2015

A small number of studies have investigated student understanding of vector addition and subtraction in generic or introductory physics contexts, but in almost all cases the questions posed were in the vector arrow representation. In a series of experiments involving over 1000 students and several semesters, we investigated student understanding of vector addition and subtraction in both the arrow and algebraic notation (usingî,ĵ,k) in generic mathematical and physics contexts. First, we replicated a number of previous findings of student difficulties in the arrow format and discovered several additional difficulties, including the finding that different relative arrow orientations can prompt different solution paths and different kinds of mistakes, which suggests that students need to practice with a variety of relative orientations. Most importantly, we found that average performance in the ijk format was typically excellent and often much better than performance in the arrow format in either the generic or physics contexts. Further, while we find that the arrow format tends to prompt students to a more physically intuitive solution path, we also find that, when prompted, student solutions in the ijk format also display significant physical insights into the problem. We also find a hierarchy in correct answering between the two formats, with correct answering in the ijk format being more fundamental than for the arrow format. Overall, the results suggest that many student difficulties with these simple vector problems lie with the arrow representation itself. For instruction, these results imply that introducing the ijk notation (or some equivalent) with the arrow notation concurrently may be a very useful way to improve student performance as well as help students to learn physics concepts involving vector addition and subtraction.

the mathematical constructions is a world of concepts within the human mind, an abstract world, separate from the outside world, the world of experience, these two worlds interact with each other through the senses and experiences they produce (measurements). This creates the miracle of understanding that Einstein said " the most incomplehensible thing about the world is that it is complehensible"