HOW TO FIND THE PERPENDICULAR DISTANCE BETWEEN TWO SKEW LINES AND PARALLEL LINES

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The Shortest Distance Between Skew Lines

The shortest distance between the lines is the distance which is perpendicular to both the lines given as compared to any other lines that joins these two skew lines.

Vector Form

We shall consider two skew lines  L1 and L2 and  we are to calculate the distance between them. The equations of the given lines are:

Here  vector  and vector  are the vectors through which line (1) and (2) passes and and   are the vectors which are parallel to lines    L1 and L2  respectively. 

Then perform the following steps

(1)                 Calculate  -  

(2 )                Calculate   ×

(3)                Calculate    |  ×  | 

(4 )                Calculate ( − ) . (   ×  )

and if the dot product of these two vectors come out to be negative then take its absolute value as distance can not be a negative quantity  .

(5 )                Put all these values in the formula given below and  the value so calculated is the shortest distance between two skew Lines.

 ⃗⃗

Problem : How to find  the shortest distance between two skew lines in vector form whose equations are given by

  Now write the values of    ,, and 

Now find out the difference of and 

After solving this determinant and little simplification we get ,

The magnitude of this vector 

                             = √(169+64+4)=√(237)

Now find Dot Product    ( -  )  and  (   ×  ) ,

                                                    = (0)(-13)+(-6)(8)+(5)(-2)

                                                    = -48 -10

                                                    = -58

Taking the absolute value of

|   (− )  .  (   ×  ) | = 58

Now putting all these values in SD Formula written above , we can have

SD = 58/(√237) Units

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The Shortest Distance Between Parallel Lines

Consider two parallel lines

Here  vector  and  vector  are the vectors through which line (1) and (2) passes and    is the vector which is parallel to both lines    L1 and L2  respectively. Now perform the following steps 

( 1 )                 Calculate  -  

( 2 )                Calculate    | | 

( 3 )                Calculate ( -  )     ×  ,

     

Put all these values in the formula given below and the value so calculated is the shortest distance between two Parallel Lines, and if it comes to be negative then take its absolute value as distance can not be negative

Problem 2 : How to find  the shortest distance between two Parallel lines in vector form whose equations are given by

As we know these are parallel line because both these equations are parallel to same vector  -2i +3J+5k.

Now write the values of    ,, and  as follows

Now find out the difference of and 

Now  Calculate ( -  )     ×  ,

     After solving this determinant and  simplification we get ,            

                  

 

The magnitude of this vector is   

= √2069

Similarly the magnitude of vector    is √38

  Put all these values in the formula given below and the value so calculated is the shortest distance between two Parallel Lines, and if it comes to be negative then take its absolute value as distance can not be negative

SD = √(2069 /38) Units

At Last

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Read previous Post How to find slope of line ax+by= c
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HOW TO FIND THE SLOPE OF LINE AX + BY = C , SLOPE OF LINE

How to find the slope of line when its equation is given,ax+by=c calculator, ax+by+c=0, meaning,ax+by+c=0 solve for y, ax+by=c given two points, ax+by=c what is c, slope formula, ax+by=c meaning,slope of a line formula, slope of a line calculator, how to find the slope of a graph,how to find slope from an equation, slope of a line definition,slope formula example,slope definition,slope of a vertical line

How To find the slope of ax +by = c

Given Equation is ax + by = c

Transferring the 1st term  containing ‘x’ to R H S

 by = c - ax 

Dividing by b to find value of  ‘y’

y = c/b - ax/b

y =  -ax/b  + c/b, 

Cancelling the terms which are going to be cancelled

Rewriting the equation compatible to y = mx+c

we get , y = (-a/b)x +c/b

Compare this equation with y = mx+c

The slope of the given equation " ax + by = c " is    m = -a/b

Hence  the slope of given line is -a/b.

Note :-So from this method we can say that slop of any line can be written as -(co eff of x /co eff of y)

How To find the slope of  x- Axis

As we know that ,The equation of X-axis is y=0

Rewriting this equation in standard form of  y = mx+c

y = 0.x + 0,

Comparing it with standard form to get m = 0,

⇒ The slope of x-axis is 0 (Zero)

How To find the slope of  Y-Axis

As we know that ,The equation of X-axis is x = 0

Rewriting this equation in standard form of  y = mx+c

0.y = 1.x + 0,

Comparing it with standard form to get m = 0,

⇒ The slope of y-axis is 0 (Zero)

How To find the slope of 4x +3y = 10

Given Equation is 4x + 3y = 10

Transforming the 1st term which contains ‘x’ to R H S

 3y = 10 - 4x

Dividing by 3 to find value of  ‘y’

   y = 10/3 - 4x/3

⇒  y =  - 4x/3  + 10/3

Rewriting the equation compatible to y = mx+c

we get , y = (-4/3)x +10/3

Comparing this equation with y = mx+c

The slope of the given equation is  m = -4/3

Hence  the slope of given line is -4/3

Note simply by applying the formula ,we can calculate the slope of  this line  -(co eff of x/co eff of y) = -4/3

How To find the slope of 2x -7y = -5

Given Equation is 2x - 7y = -5

Transforming the 1st term  containing ‘x’ to R H S

- 7y = -5 - 2x 

Dividing by -7 to find value of  ‘y’

-7y/(-7) = -5/(-7) - (2/-7)x ,

Cancelling  -ve sign of the num with -ve sign of den,we have

⇒  y =  2x/7  + 5/7

Comparing the above  equation  with y = mx+c

we get , y = (2/7)x +5/7,

The co eff of  'x' on the right hand side is the value of slope

The slope of the given equation is  m = 2/7

Hence  the slope of given line is 2/7

Simply by applying the formula ,we can calculate the slope of  this line  -(co eff of x/co eff of y) = -(-2)/(-7) = 2/7

How To find the slope of √2x +√5y = 5

Given Equation is √2x +√5y = 3

Transforming the 1st term  containing ‘x’ to R H S

√5y = 3 - √2x 

Dividing by √5 to find value of  ‘y’

√5y/(√5) = 3/(√5) - (√2/√5)x ,

Cancelling  -ve sign of the num with -ve sign of den,we have

⇒  y =  - (√2/√5)x +3/(√5)

Comparing the above  equation  with y = mx+c

we get , y = - (√2/√5)x +2/7,

The co eff of  'x' on the right hand side is the value of slope

The slope of the given equation is  m =  - (√2/√5)

Hence  the slope of given line is  - (√2/√5).

Thanks for devoting your precious time to this post How to find the slope of line when its equation is given,ax+by=c calculator, ax+by+c=0, meaning,ax+by+c=0 solve for y, ax+by=c given two points, ax+by=c what is c, slope formula,ax+by=c meaning,slope of a line formula,

Also Read this post How to find distance between two points in plane  and in space  

HOW TO INTEGRATE INTEGRAL WITH SQUARE ROOT IN NUMERATOR

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HOW TO INTEGRATE, INTEGRAL WITH SQUARE ROOT IN NUMERATOR

Let us find out the  Integration of quadratic polynomial  in denominator, integral of square root of polynomial, integrals with square roots in denominator, integration of linear by quadratic, integral with square root in numerator ,integration of square root formula .

Click here to Watch more  Integration Videos

Let us take some examples to integrate such type of example.

Problem

1st of all making the co eff off  x²  positive and unity if it is not and then    rewrite  x² +4x -1 as sum/ difference 
of  (a + b )²  = a² +b² 土2ab  

and c²  as follows :- 

Now write the remaining term to equalise  x² + 4x -1,

Now writing the last term as square root of term and then multiplying the co eff of x²  with -ve sign which was earlier taken outside  .

Therefore (1) can be written as 

Now use the formula written below 

Here in this case 'x' will be replaced by 'x + 2' and 'a' will be replaced by √5

Where 'c' is called constant of integration.

This video will explains very easy and short method of such types of integration

Problem

1st of all making the co eff off  x²  positive and unity if it is not and then rewrite  2x² + x - 1 as sum / difference of 
 (a + b )²  = a² +b² 土2ab  

and c²  as follows :- 

2x² + x -1 =2[ x² +(1/2) x - (1/2)  ]
2x² + x -1 = 2[  (x )² + 2× (1/4)(x) +(1/4)² - (1/4)²  -1/2]

2x² + x -1 =  2[ {(x)² + 2 (1/4)(x) +(1/4)²}- {(1/4)²  +1/2}]

                =  2[{(x) + (1/4)}² - {1/16+1/2}]

                 = 2 [{(x) + (1/4)}² - {9/16}]

                 = 2[{(x) + (1/4)}² -  {3/4}² ]

Now using the Result 

Now replacing x by x + (1/4) and 'a' by 3/4 ,
The integration of this Integrand can be written as follows

Where 'c' is called constant of integration.

Now multiplying with 2 all the terms and After simplification we shall have 

In this post we discussed Integration of quadratic polynomial  in denominator,integral of   square root of polynomial,integrals with square roots in denominator, integration of linear by quadratic, integral with square root in numerator ,integration of square root formula,integration of square root formula,integral of root 2, anti derivative rules for square roots Please like it and  share this post with yours friends.

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