THE DIRECTION THAT DOES NOT CHANGE DIRECTION

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The article has some math. But it will be fun and exciting.

Consider the point P(2,3) in the graph. It has x-coordinate 2 and Y-coordinate 3. We can draw an arrow mark from the origin to the point and call it a vector and represent it by (2,3).
Multiply the vector by this matrix.

The product yields a vector (-3,2). Mark it in the graph. Note, the original vector is rotated by 90 degrees. A matrix can rotate, shear, scale, unshape a vector, or a point in the graph. Hence this technique is widely used in computer graphics.
Now consider the vector (0,0) or origin and multiply it by the same matrix.
[0 -1 * [0 = [0
1 0] 0] 0]

The vector is not distributed. That is, the origin cannot be rotated. It is common sense but an important point. We call this vector (0,0) as "eigenvector' of matrix [0 -1],[1 0].

"The eigenvector is one the direction of which cannot be changed by that matrix".The eigenvector of the matrix
[-6 3 is [1
4 5] 4]
If you matrix-multiply both, you get(6, 24). If you simply multiply (1,4) by 6, you get again (6, 24). It confirms that the vector is simply 'scaled six times' but its direction remains the same. You can mark them in a graph and find it is true.

The value 6 is called 'eigenvalue'. The eigenvalue is the scaling factor.

FINDING EIGEN VECTOR:
Also verifying the above case:

Let (x,y) be an eigenvector of our matrix (-6,3
4,5). 6 is the eigenvalue.
The principle:
Matrix *eigen vector = Eigen value * Eigen vector

Hence, (-6 3 *(x = 6*(x
4 5) y) y)
Matrix-multiplying, we get 2 equations:
-6x+3y =6x
4x +5y =6y
Rearranging:
-12x +3y =0
4x-1y =0
Solving we get, x =1 and y =4
or the eigenvector is (1,4), We got it! The eigenvector principle vector is widely used in google search algorithms, computer graphics, statistics, etc.

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LISSAJOUS FIGURES

Definition: "When a particle is subjected to two sine wave motion or two oscillatory motion at right angles, the particle describes lissajous figures". We know sine wave motion and circular motion is basically same. Hence we draw two circles A and B perpendicular to each other. The circle B rotates twice faster than circle A. That is, frequency of circle B is two times than that of A. A particle at the intersection of two circles is subjected to two sine wave motion A and B at 90 degree simultaneously. The particle will describe figures depending on the frequency and phase of A and B . In our case, the ratio of frequency is 1:2 and the two waves are in phase. To draw lissajous figures : A moving point in both the circles are chosen. Here we should remember; during the time taken by the circle A to complete one rotation, circle B completes two. Hence the points are marked on the circles according to their speed. Then straight lines

THE PARABOLA

A jet of water shooting from a hose pipe will follow a parabolic path. What is the so special about parabola. Y= x^2 Draw a graph for the above equation. It will result in a parabola. This parabola is also called unit parabola. Any equation involving square will yield a parabola. Example: Y = 2x^2 +3x+3 (also called quadratic equation) X= 2 and -2, both satisfies the equation 4 = X^2. Parabolic equations always have two solutions. Any motion taking place freely under gravity follows parabolic path. Examples: An object dropped from a moving train, A bomb dropped from flying plane, A ball kicked upwards. If a beam of light rays fall on the parabolic shaped mirror, they will be reflected and brought to focus on a point. This fact is made use of in Dish Antenna, Telescope mirrors, etc. Inverted parabola shape is used in the construction of buildings and bridges. Because the shape is able to bear more weight. A plane

CASINO'S GAME

Let us find out how the casino survives with mathematics. Say, your friend invite you for a game of dice. You must bet (wager) 2 dollars. If you roll 'six' you will get back 8 dollars. The game will go on for 30 rounds. All sounds good. The probability of rolling 'six' is 1/6. Since the game will be played for 30 times, the 'expected win' is 30*1/6 = 5. That is, you are expected to win 5 rounds out of 30. Hence your gain will be 5 * 8 =40 dollars. ok. This also implies that you will loose 25 rounds. Hence your loss will be 25*2 =50 dollars. Your net gain will be gain-less = 40-50 = -10 dollars. For 30 rounds, the loss is -10 dollars, Hence, for one round =-10/30 = -1/3 dollars. There will be a loss of -1/3 or 0.33 dollars per round. It is not a fair game. Let us make a simple formula to calculate 'Pay out per round\. The probability for a win = p The pay-out in case of win = V No. of rounds = n The expect

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