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Hey im doing some functions questions and im looking at my notes and i cant figure out how to do them. I also have answered the first 3 and i would appreciate if anyone could tell me if i am correct...

1.  Let $p_1$ = (-12, 10) and $p2$ = (14,-10). Find:
i.  the Euclidean distance between p¬1 and p2 = 32.80
ii. the Manhattan distance between p¬1 and p2 = 46
iii.    the mid-point between p¬1 and p2 (1,0)

2.  Let $p1$ = (5, -7, 4) and $p2$ = (3, 6, -4). Find:
i.  the Euclidean distance between p¬1 and p2 = 69
ii. the Manhattan distance between p¬1 and p2 = 23
iii.    the mid-point between p¬1 and p2 = 

3.  Which of the following are equations of lines:
i.  2 x + 5 y = -2  True
ii. –5.3 – 9.7 x2 + 4 y = -3 False
iii.    y = 0.75 x – 1.4  True

4.  What is the slope of each of the lines in question 4?

5.  The slope of a line that passes through the point (2,3) is 4. What is the equation of the line?

6.  Find an equation of the line that is perpendicular to the line -2 x - 6 y = -18.4 and passes 
through the point (4,-6).

 7.  Determine the equation of the curve of shortest distance which joins the point (2,-1) to the 
      line 2x-5y= - 4. 
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What is p¬1? $p_1$? Also, I noticed you originally had this typed up as opposed to linking a picture, which is usually preferred. To get your layout as you'd like it, note (i) add a double space at the end of a line to force a line break; and (ii) most basic math is rendered in math font when putting it between dollar signs. For subscripts, use x_1 between dollars. –  gnometorule Feb 13 '13 at 20:57
Thank you. I am new to the mathematics section of stack overflow and i appreciate the comments. –  Pendo826 Feb 13 '13 at 21:00
The answer to 2.(i) is about $15.39$. The missing midpoint is $(4,-1/2,0)$. –  André Nicolas Feb 13 '13 at 21:01
Your indentation at the beginning of the line is preventing the MathJax from taking effect. Also, I think it's bad form to just copy out questions from a book, especially multiple in a single question. Try to answer them yourself, and when you get stuck, ask a question about the concept you don't understand. –  Ben Millwood Feb 13 '13 at 21:44

1 Answer 1

up vote 1 down vote accepted
    1. Euclidian distance is $\sqrt {(14 - (-12))^2 + (-10-10)^2} = \sqrt{26^2+20^2}\approx32.80$, so you're right.
    2. Manhattan distance is $\left|14-(-12)\right|+\left|-10-10\right| = 46$, so you're right there, too.
    3. Midpoint is $(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}) = (\frac{14-12} {2},\frac{-10+10}{2})=(1,0)$, so that's also correct.
    1. $\sqrt{(5-3)^2+(-7-6)^2+(4-(-4))^2}=\sqrt{237}\approx15.395$
    2. 23 is right.
    3. The midpoint is $(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2}) =(\frac{5+3}{2},\frac{-7+6}{2},\frac{4-4}{2})=(4,\frac12,0) $
  1. Your answers are correct.
  2. Your question makes no sense; no lines are given in number 4.
  3. Since you have a point and the slope of the line, you can use point-slope form: $$ \begin{align} y-y_0 &= m(x-x_0)\\ y-3&=4(x-2)\\ y&=4x-5 \end{align} $$
  4. The slope of the given line is $-\frac13$, so the slope of the perpendicular line is the opposite reciprocal. That is, $m=3$. So, using the point-slope form again, $$ y-(-6)=3(x-4)\\ y=3x-18 $$
  5. If you're just asking for the shortest distance to the line, you can use the formula $$ \frac{\left|Ax_1+By_1+C\right|}{\sqrt{A^2+B^2}}, $$ which gives $\frac{13}{\sqrt{29}}\approx2.41$
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Brilliant, Thanks for answering those question. That has really helped alot. :) –  Pendo826 Feb 13 '13 at 21:28

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