Showing posts with label Maths. Show all posts
Showing posts with label Maths. Show all posts

How to find proportion of Four numbers

If A : B = 2 : 3, B : C = 4 : 5 and      C : D = 6 : 7,    what is A  : D  ?

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HOW TO FIND SHORTEST DISTANCE BETWEEN TWO LINES

SHORTEST DISTANCE BETWEEN TWO LINES

1st of all we shall find out shortest distance between two Parallel lines.

Problem 1

Consider two parallel lines whose equations in vector form are given by
HOW TO FIND SHORTEST DISTANCE BETWEEN TWO LINES

 Now comparing these equations with standard form , and write

 , and  vectors ,we get
HOW TO FIND SHORTEST DISTANCE BETWEEN TWO LINES

Now applying this formula to find the shortest distance between two lines .
HOW TO FIND SHORTEST DISTANCE BETWEEN TWO LINES
As it is clear from formula , we have to find cross product of  and  and then magnitude of vector 

HOW TO FIND SHORTEST DISTANCE BETWEEN TWO LINES

Now find the magnitude of  × vector
HOW TO FIND SHORTEST DISTANCE BETWEEN TWO LINES
                                 =√(81)+(196)+(16)
                                 =√293
Magnitude of 
HOW TO FIND SHORTEST DISTANCE BETWEEN TWO LINES
                    = √49
           = 7
Putting all these values in  the formula of Shortest Distance between two lines .
HOW TO FIND SHORTEST DISTANCE BETWEEN TWO LINES
Now Find distance between two skew lines i.e. Lines which are not Parallel lines.

Problem 2

Consider two parallel lines whose equations in vector form are given by
 Now comparing these equations with standard form , and write

 , and  vectors ,we get
 Now applying this formula to find the shortest distance between two lines 

 Now find cross product and then  magnitude of these two vectors


Don't Forget Watch this Video to understand better


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Find two positive numbers whose sum is 16 and sum of whose cube is Minimum


Show that of all the rectangles inscribed in a circle of given radius . The Square has maximum Area.


Solutions


Let ABCD be rectangle which is  inscribed in a given circle of radius ‘r’
Show that of all the rectangles inscribed in a circle of given radius . The Square has maximum Area.
And Let θ be the angle between side of rectangle and Diameter of given circle.


Therefore from right angled  Δ ABC ,

We have 
  AB  = AC cosθ          ∵ AC = 2r
Let A(x) be the area of Rectangle ABCD
∴ A(x) = AB × BC
    A = (2r cos θ)(2r sin θ )
    A =  4r2 sin θ cos θ
    A = 2r2  (2sin θ cos θ)
    A = 2r2  (sin 2θ )

⇒ 2r2 2 (cos 2θ ) = 0 ,As r2 is constant
⇒cos 2θ = 0
⇒cos 2θ =cos (π/2)
⇒ θ = π/4
 =4r2  (-2sin 2θ 



∴ A has Maximum value at θ = π/4



Find two positive numbers whose sum is 16 and sum of whose cube is Minimum

Solution

Let us consider two numbers x and 16- x .
Then transforming our problem to mathematical form which says “sum of whose cube”  as follows
A (x) =   x3 + (16 - x)3…….. (1)
Differentiating both sides w .r. t  “x” , we get



     X = 8
So  x  =  8 will be the 1st required numbers if Double derivatives of A  w. r. t  ‘x’ comes to be positive at x = 8.
Differentiate (2)  w. r. t. ‘x’  .




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HOW TO SOLVE LINEAR EQUATIONS OF THREE VARIABLES BY MATRIX METHOD

How to find solutions of system of linear equations of three variables with  the help of Matrix Method. Let us solve three linear equations of three variables given below by Matrix Method.


HOW TO SOLVE LINEAR EQUATIONS OF THREE VARIABLES BY  MATRIX  METHOD

These are not simple linear equations to solve. But these equations can be made simple by following substitution .



After substitution , these equations are reduced to following simple linear equations
2x + 3y + 4z = 3
3x - 4y + 5z = 5
  x + 2y - 3z = 6
To solve these equations by Matrix Method , Let us transform theses set of linear equations to Matrix form

AX = B,   and  =A-1 B---------------- (4)

where A is matrix comprises the coefficients  x , y and z respectively as shown in the matrix given below , And B is the matrix consist of constant terms on right hand side of these equations and X is the matrix of variables in the linear equations.
HOW TO SOLVE LINEAR EQUATIONS OF THREE VARIABLES BY  MATRIX  METHOD

1st of all we have to find A- and then product of the matrices A-1 and  as mentioned in equation (4). But to find inverse of matrix A, its determinant value must be non zero   ( Note it) .
Let us find determinant value of Matrix A as follows :-

|A| = 2[(-4)(-3)–(5)(2)]-3[(3)(-3)-(5)(1)]+4[(3)(2)-(-4)(1)] 

|A| = 2[12-10]-3[-9-5]+4[6+4] 
|A| = 2[2] -3[-14]+4[10] 
|A| = 4 +42+40 
|A| = 86 
Since the value of Determinant of Matrix a is non zero.
  ⇒    A-1  exists .
To find inverse ,we have to find ad joint of Matrix A . And to find Ad joint of any Matrix , we have to find Co factor Matrix . 

What is Co Factor Matrix


Co factor Matrix can be obtained by the co factors of all the elements of Matrix written in same place  where the element was written originally. 
Formula for Co factor of any element 
        Cij = (-1)i+jMij 


Co factor of A11 

Co factor of A11   element can be calculated by eliminating 1st row and 1st column and solving the remaining determinant.
(-1)1+1M11 
= 12 - 10 = 2


Co factor of A12 


Co factor of A12   element can be calculated   by eliminating 1st row and 2nd column and solving the remaining determinant.
(-1)1+2M12 


= -1(-9 - 5 )= 14

Co factor of A13 


Co factor of A13   element  can be  calculated   by eliminating 1st row and 3rd column and solving the remaining determinant.
(-1)1+3M13 
= 6 + 4 = 10


Co factor of A21 


Co factor of A21   element can be calculated   by eliminating 2nd row and 1st column and solving the remaining determinant.
(-1)2+1M11 

= -1(- 9 - 8) = 17



Co factor of A22 


Co factor of A22   element  can be calculated   by eliminating 2nd row and 2nd column and solving the remaining determinant.
(-1)2+2M22 
= - 6 - 4 = -10


Co factor of A23 


Co factor of A23   element can be calculated   by eliminating 2nd row and 3rd column and solving the remaining determinant.

(-1)2+3M23 
= -1(4 - 3) = -1

Co factor of A31


Co factor of A31   element can be calculated   by eliminating 3rd row and 1st column and solving the remaining determinant.

(-1)3+1M31 
 
= 15 + 16 = 31


Co factor of A32



Co factor of A32   element  can  be calculated   by eliminating 3rd row and 2nd column and solving the remaining determinant.

(-1)3+2M32 
= -1(10 - 12) = 2


Co factor of A33



Co factor of A33   element  can  be calculated   by eliminating 3rd row and 3rd column and solving the remaining determinant.


(-1)3+3M33 
= - 8 - 9 = -17


Co Factor Matrix of A


Now write all the co factors calculated in Matrix Form as follows
HOW TO SOLVE LINEAR EQUATIONS OF THREE VARIABLES BY  MATRIX  METHOD

Ad joint Matrix of A


To find the Ad Joint of  Co factor Matrix, transform 1st row to 1st column, 2nd row to 2nd column  and 3rd rows to 3rd column as follows 




Formula to Find   A-1 



To Find the value of  A-1 ,Putting the value of Adj A in the formula given below.
HOW TO SOLVE LINEAR EQUATIONS OF THREE VARIABLES BY  MATRIX  METHOD


Find values of x ,y and z


Multiplying both the matrices which are on the right side.
HOW TO SOLVE LINEAR EQUATIONS OF THREE VARIABLES BY  MATRIX  METHOD
Simplifying the matrix so obtained
HOW TO SOLVE LINEAR EQUATIONS OF THREE VARIABLES BY  MATRIX  METHOD
Dividing each element by 86 to get matrix of 3×1 (i. e. Perform scalar multiplication of matrix ).
HOW TO SOLVE LINEAR EQUATIONS OF THREE VARIABLES BY  MATRIX  METHOD

Using the equality of two matrices .We can write the values of x, y  and z respectively like this

 x =277/86  , y = 4/86  and z = -77/86

Now the values of P, Q and R can be calculated by putting the values of x , y and z in equations (1) , (2 ) and (3) respectively.

P = 86/277  ,Q = 86/4   and R = -86/77

Don't Forget to Watch this Video of same Method

Verification of solution



Putting values of P , Q and R in given  equations , these values must satisfies  three given equations
From 1st equation

HOW TO SOLVE LINEAR EQUATIONS OF THREE VARIABLES BY  MATRIX  METHOD

Similarly other two equations can be verified

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Conclusion


I have discussed the method of solving the System of linear Equations with the help of Matrix Method , method to find inverse of a matrix ,How to find inverse of 3×3 Matrix,how to solve determinant,how to solve system of equation by matrix,matrix method of solving system of equations of three variables,If you liked the post Don't  forget to share it with your friends , And in case of any improvement please make use of Comment Box . 

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