Find the distance between lines 8x−15y+5=0 and 16x−30y−12=0.

The distance between the two parallel lines is 11/17 units

Step-by-step explanation:

we have

8x-15y+5=0 -----> equation A

16x-30y-12=0 ----> equation B

Divide by 2 both sides equation B  

8x-15y-6=0 ----> equation C

Compare equation A and equation C

Line A and Line C are parallel lines with different y-intercept

step 1

Find the slope of the parallel lines (The slope of two parallel lines is the same)

8x-15y+5=0

15y=8x+5

the slope is

step 2

Find the slope of a line perpendicular to the given lines

Remember that

If two lines are perpendicular then their slopes are opposite reciprocal (the product of their slopes is -1)

we have

therefore

step 3

Find the equation of the line perpendicular to the given lines

assume any point that lie on line A

For

To find the equation of the line we have

 ---> is the y-intercept

The equation in slope intercept form is

-----> equation D

step 4

Find the intersection point of the perpendicular line with the Line C

we have the system of equations

----> equation D

----> ----> equation E

equate equation D and equation E and solve for x

Multiply by 120 both sides to remove fractions

Find the value of y

the intersection point is

step 5

Find the distance between the points and

the formula to calculate the distance between two points is equal to

substitute the values

Simplify

therefore

The distance between the two parallel lines is 11/17 units

see the attached figure to better understand the problem

the value of v.w= -12.

step-by-step explanation:

we are given three vector r,v and w as:

r = < 5, -5, -2> ; v = < 2, -8, -8> ; w = < -2, 6, -5>

we know that r,v and w could also be represented as:

r=5i-5j-2k

v=2i-8j-8k

and w= -2i+6j-5k

where i,j and k are the unit vectors of x,y and z-axis respectively.

also we know that it has the following property:

i.i=1 ; j.j=1; k.k=1

and i.=j.i=i.k=k.i=j.k=k.j=0

i.e. when we multiply two vectors or we find it's dot product we need to only multiply the components of x coordinates with each other; y with each other and z with each other.

hence, v.w= (2i-8j- -2i+6j-5k)

                  = 2×(-2)+(-8)×(6)+(-8)×(-5)

                  = -4-48+40

                  = -12

hence, v.w= -12