1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
mina [271]
3 years ago
11

How to find the smallest distance between two lines?

Mathematics
1 answer:
Lostsunrise [7]3 years ago
4 0
Consider two lines in space `1 and `2 such that `1 passes through point P1 and is parallel to vector ~v1 and `2 passes through P2 and is parallel to ~v2. We want to compute the smallest distance D between the two lines.
If the two lines intersect, then it is clear that D = 0. If they do not intersect and are parallel, then D corresponds to the distance between point P2 and line `1 and is given by D = k−−−→ P1P2 ×~v1k k~v1k . Assume the lines are not parallel and do not intersect (skew lines) and let ~n = ~v1 ×~v2 be a vector perpendicular to both lines. The norm of the projection of vector −−−→ P1P2 over ~n will give us D, i.e., D = |−−−→ P1P2 ·~n| k~nk . Example Consider the two lines `1 : x = 0, y =−t, z = t and `2 : x = 1+2s, y = s, z =−3s. It is easy to see that the two lines are skew. Let P1 = (0,0,0), ~v1 = (0,−1,1), P2 = (1,0,0), and ~v2 = (2,1,−3). Then, −−−→ P1P2 = (1,0,0) and ~n = ~v1 ×~v2 = (2,2,2). We then get D = |−−−→ P1P2 ·~n| k~nk = 1 √3. Observe that the problem can also by solved with Calculus. Consider the problem of minimizing the Euclidean distance between two points on `1 and `2. Let Q1 = (x1,y1,z1) and Q2 = (x2,y2,z2) be arbitrary points on `1 and `2, and let F(s,t) = (x2 −x1)2 +(y2 −y1)2 +(z2 −z1)2 = (1+2s)2 +(s + t)2 +(−3s−t)2 = 14s2 +2t2 +8st +4s+1. Note that F(s,t) corresponds to the square of the Euclidean distance between Q1 and Q2. Let’s nd the critical points of F. Fs(s,t) = 28s+8t +4 = 0 Ft(s,t) = 4t +8s = 0 By solving the linear system, we nd that the unique critical point is (s0,t0) = (−1/3,2/3). Since the Hessian matrix of F, H =Fss Fst Fts Ftt=28 8 8 4, is positive denite, the critical point corresponds to the absolute minimum of F over all (s,t)∈R2. The minimal distance between the two lines is then D =pF(s0,t0) = 1 √3.
You might be interested in
Carlos used 6 boards to build 2/3 of a front porch. How many boards will Carlos use to build the whole porch? Express your answe
antiseptic1488 [7]

Answer:

9 boards

Step-by-step explanation:

2/3 of the porch= 6 boards

so to figure out how much one third of the porch needs you find half of what it takes to fill in 2/3 of the porch and you get 3.

so 3=1/3 of the porch

then you simply do 3times 3 or 3+3+3 and you get 9

so it takes 9 boards to fill the whole porch

8 0
3 years ago
Find the missing term of the following sequence. . . . –2, __, 456, . . .
masha68 [24]

The answer should be

228

3 0
3 years ago
Read 2 more answers
If the simple interest on $2,000 for 3 years is $300 then what is the interest rate?
Alona [7]

Answer:

100 dollars a year

Step-by-step explanation:

4 0
2 years ago
How do you solve multiplying binomials with example: (x+4)^2
salantis [7]
This can be written as (x+4)(x+4)

Multiply everything in the second parenthesis by the x.

x^2 + 4x

Multiply everything in the second parenthesis by 4.

4x + 16

Add these two equations together.

x^2 + 4x + 4x + 16

Combine like terms.

x^2 + 8x + 16

Hope this helps!
8 0
3 years ago
A local high school held a two-day fundraiser selling lunches. The equations shown represent the amount of money the school coll
scoundrel [369]

The given equations are

12x + 4y = 152 ...........1

32x + 12y = 420 .....2

Multiplying equation 1 by 3 to make  coefficient of y same we have:

36x+12y= 456

32x+12y=420

Subtracting the two equations we have:

4x=36.

Dividing by 4 both sides

x= 9.

Substituting x value in equation 1

12(9) + 4y = 152

108+4y=152

Subtracting 108 both sides: 4y=44.

Dividing both sides by 4.

y=11.

cost of the vegetarian lunch is $11.


In the equations, x represents the cost of a chicken lunch and y represents the cost of a vegetarian lunch.




5 0
3 years ago
Other questions:
  • Find the difference between the product of 26.22 and 3.09 and the sum of 3.507, 2.08, 11.5, and 16.712.
    12·2 answers
  • Solomon is 9 years younger than his brother.His brother is 21.How old is Solomon.write and solve an equation
    12·2 answers
  • X + 2y = 5
    15·2 answers
  • Determine the surface of the cover of a textbook that has a length of 11 inches, a width of 8 inches, and a height of 3 inches.
    13·1 answer
  • Please help me this is due tmrw
    12·1 answer
  • Solve for x, given the equation Square root of x plus 9 − 4 = 1.
    13·1 answer
  • How can you solve the equation 4x= 2X-2 graphically?
    5·2 answers
  • Which of the following expressions could be used to find the sale price of an item that costs d dollars if it is on sale for 30%
    8·1 answer
  • When f(x) =-3 what is x?
    8·2 answers
  • What’s the equation to this table?Pls help.Use y=mx+b
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!