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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.
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Akimi4 [234]

Multiply the demand values by the weights and report the sum.

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The weighted moving average forecast for the 5th period is 3,760.

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3 years ago
What is the equivalent fraction of 12/15 with a numerator of 12​
Ratling [72]

Answer:

Step-by-step explanation:

12/15 has a numerator of 12 already.

Some equivalent fractions are 4/5 and 24/30.

5 0
3 years ago
Here is a circle.
Stolb23 [73]
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3 years ago
A teacher gave a test to a class in which 10% of the students are juniors and 90% are seniors. The average score on the test was
IceJOKER [234]

Each junior  receives 93 score on the test. It is also the average score of the juniors.

Given Information

It is given that 10% of the strength of the class consists of juniors and 90% consists of seniors.

Let us assume the total strength of the class is 100, then,

Number of seniors = 90

Number of juniors = 10

It is also given that the average score of the class = 84

And, average score of the seniors = 83

Another given information is that all the juniors all received the same score, which is their average score. Let it be x.

Average Score of Juniors

Total marks obtained by the seniors = Number of seniors × Average score of seniors

= 90 × 83

= 7470

Total marks obtained by the juniors = Number of juniors × Average score of juniors

= 10x

Average marks on the test = Total marks / Total number of students

⇒ 84 = (7470 + 10x)/100

⇒ 84 × 100 = 7470 + 10x

⇒ 10x + 7470 = 8400

⇒ 10x = 8400-7470

⇒ 10x = 930

⇒ x = 930/10

⇒  x = 93

Thus, each junior scores 93 on the test.

Learn more about average here:

brainly.com/question/24057012

#SPJ4

5 0
2 years ago
Graph the system below and write its solution.
Diano4ka-milaya [45]

Answer:

See the graph attached. It has one solution: (6,-4)

Step-by-step explanation:

The slope-intercept form of a line is:

y=mx+b

Where m is the slope and b is the intersection of the line with the y-axis.

Given the first equation  y =\frac{-1}{2}x -1

You can identify that:

b=-1

Substitute y=0 to find the intersection with the x-axis

0 =\frac{-1}{2}x -1\\1(-2)=x\\x=-2

This line passes through the points (0,-1) and (-2,0)

Given the second equation:

-2 + y = -6

Solve for y:

y = -6+2\\y=-4

It passes through the point (0,-4).

Now, you can graph. See the figure attached.

It has one solution,which is the point of intersection of both lines:  (6,-4)

5 0
3 years ago
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