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

 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 .

As it is clear from formula , we have to find cross product of  and  and then magnitude of vector 

Now find the magnitude of  × vector

                                 =√(81)+(196)+(16)

                                 =√293

Magnitude of 

                    = √49

           = 7

Putting all these values in  the formula of 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

FINAL WORDS

Thanks for watching and responding to this post "HOW TO FIND SHORTEST DISTANCE BETWEEN TWO LINES".

My previous Posts

HOW TO SOLVE LINEAR EQUATIONS OF THREE  VARIABLES  BY  MATRIX  METHOD

 How to find area bounded by two circles 

 How to integrate integral with square root in numerator     

How to find area of the circles which is interior to the parabola
How to find common area of two parabolas.

Appeal

If you are a mathematician Don't forget to visit my Mathematics You tube channel , Mathematics Website and Mathematics Facebook Page , whose links are given below

Maths Website , Facebook Page and You Tube Channel

Read More

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’

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

My previous Posts

HOW TO SOLVE LINEAR EQUATIONS OF THREE  VARIABLES  BY  MATRIX  METHOD

 How to find area bounded by two circles 

 How to integrate integral with square root in numerator     

How to find area of the circles which is interior to the parabola

How to find common area of two parabolas.

FINAL WORDS

Thanks for watching and responding to this quiz based on Matrices and Determinants on Mathematics. 

Appeal 

If you are a mathematician Don't forget to visit my Mathematics You tube channel ,Mathematics Website and Mathematics Facebook Page , whose links are given below

Maths Website , Facebook Page and You Tube Channel

Read More

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. In this post we shall  solve such types of system of linear equations in which variables lies in the denominators.  Let us solve three linear equations of three variables given below 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 can be  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 this set of linear equations to Matrix form

AX = B, and  X =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.

1st of all we have to find A^{-1  and then product of the matrices} A^-1 and B  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 

        C_ij = (-1)^i+jM_ij 

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

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.

Find values of x ,y and z

Multiplying both the matrices which are on the right side.

Simplifying the matrix so obtained

Dividing each element by 86 to get matrix of 3×1 (i. e. Perform scalar multiplication of matrix ).

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

Similarly other two equations can be verified

TO CRACK  ANY COMPETITIVE EXAM VISIT THIS YOUTUBE CHANNEL FOR MATHS AND REASONING  EXAM CRACKER

My Previous Posts

Don't forget to   read this posts

Quiz of  Mathematics For You 

 How to find area bounded by two circles 

 How to integrate integral with square root in numerator     

How to find area of the circles which is interior to the parabola

How to find common area of two parabolas.

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 . 

Appeal

If you are a mathematician Don't forget to visit my Mathematics You tube channel , Mathematics Website and Mathematics Facebook Page , whose links are given below

Maths Website , Facebook Page and You Tube Channel

Read More

HOW TO RECALL FORGOTTEN PIN OF ATM CARD A REASONING QUESTION

Solve this Mathematics problem in 5 minutes

A house wife forgot her 'ATM PIN' which is a four digit number, but luckily she remembers some hints on how to recall this 'PIN'

Here are some of the clues

1. The 1st digit is half of the 2nd

2. The sum of the 2nd and 3rd digits is 10

3. The 4th is equal to the 2nd plus 1

4. The sum of all digits is 23

What is her ATM PIN?

Solution

Let the four digits PIN be   wxyz, Here 1st ,2nd .3rd and 4th digit are w,x,y and z  respectively .

According  to 1st condition

The 1st digit is half of the 2nd i.e
w = x/2   ............................. (1)

 2nd condition

The sum of the 2nd and 3rd is 10 i.e

x + y = 10

This implies   y = 10 - x  .................... ( 2 )

3rd condition

 The 4th is equal to the 2nd plus 1    i . e

z = x + 1     .......................... ( 3 )

According  to 4th condition

 The sum of all digits is 23 . i e .

w  +  x + y + z  = 23 ............ ( 4 )

How to find 2nd digit

1st of all we have to find the value of x .because x is related to all other equations.

Putting the values of w ( from eq1) , y (from eq 2), z( from eq3) in ( 4) we get

x/2   + x + 10 - x + x + 1 = 23,cancelling 'x'

3x/2 +11 = 23
3x/2 =23-11
3x/2 = 12

x = 8 , This is our 2nd  digit

Now put the value of x in  (1)
w = 8/2
w = 4  , This is our 1st  digit

To find 3rd digit Put the value of x in (3)

z = x + 1  

z = 8 + 1

z = 9, This is our 4th  digit

To find 3rd digit Put the value of x in (2)

y = 10 - 8

y = 2 , This is our 3rd  digit

So The PIN Would be  wxyz  ➡️4829

Verification

1 The 1st digit is half of the 2nd      ➡️   4  and  8 , definitely 4 is half of 8

2. The sum of the 2nd and 3rd is 10  ➡️ 8+2 =10

3. The 4th is equal to the 2nd plus 1  ➡️ 9 = 8+1

4. The sum of all digits is 23   ➡️ 4 + 8 + 2 + 9 = 23

Conclusion

Thanks for visiting this website and spending your valuable time to read this post regarding How to recall forgotten  PIN of  ATM  .If you liked this post , don't forget to   share it with your friends to benefit them also ,we shall meet in next post , till then bye and take care......

If you are a mathematician Don't forget to visit my Mathematics You tube channel ,Mathematics Website and Mathematics Facebook Page , whose links are given below

Maths Website , Facebook Page and You Tube Channel

Read More