## anonymous 4 years ago For the given points P,Q.R, find the approximate measurements of angle PQR. P: (1,-4), Q: (2,7), R: (-2,2)

1. Mertsj

|dw:1328102812701:dw|

2. anonymous

have to solve it by dot products in calc 3

3. phi

Mertsj has already figured out the lengths of these vectors

4. anonymous

yea but how do you go about to dove this? i have no clue? i am trying all kinds of different things

5. anonymous

solving*

6. phi

First, let me fix it. we want PQR $(\overrightarrow P -\overrightarrow Q)\cdot (\overrightarrow R -\overrightarrow Q)= |\overrightarrow P -\overrightarrow Q||\overrightarrow R-\overrightarrow Q|\cos\theta$

7. phi

I assume you know how to find P-Q and R-Q : subtract corresponding elements I'll do P-Q (1,-4) - (2,7)= (1-2, -4-7)= (-1, -11)

8. phi

the length of a vector (x,y) = | (x,y)| = sqrt(x*x + y*y)

9. phi

a dot product of (x,y) and (a,b)= ax+by

10. phi

Did you find R-Q ?

11. anonymous

yea its (-4,-5)

12. phi

Did you find the magnitude of R-Q (and P-Q) ?

13. anonymous

you just square them right?

14. phi

the length of a vector (x,y) = | (x,y)| = sqrt(x*x + y*y) that means you take the first number , square it, square the 2nd number, add together then take the square root

15. anonymous

p-q = sqrt (121) r-q = sqrt (41)

16. phi

double check p-q

17. anonymous

oh its sqrt (122)

18. phi

ok , now the dot product of (p-q) dot (r-q) (-1,-11) dot (-4, -5)

19. phi

dot product of (x,y) and (a,b)= ax+by

20. anonymous

so its (-1*-4 + -11 * -5)

21. phi

yes, but that is just a number=?

22. anonymous

it = 59?

23. phi

yes. now put it all together (p-q) dot (r-q) = |p-q| | r-q| cos A we now everything except A.

24. anonymous

but that will only give us one angle right?

25. phi

This gives us the angle formed by p-q and r-q p-q is a vector with its base at q. same for r-q |dw:1328106371363:dw|

26. anonymous

yea but we need three angles angle P=? , angle Q=? , and angle R =?.

27. phi

find the approximate measurements of angle PQR this means the angle from P to Q to R in other words, Q is the vertex . Just 1 angle

28. anonymous

this is problem 60 in the book. problem 61 is the same question but P: (0, -1, 3), Q = (2,2,1), R= (-2, 2, 4) and answer given in the back is p=78.8 degrees , q= 47.2 degrees and r = 54.0 degree

29. anonymous

thats's why i am confused

30. phi

First, what did you get for angle pqr?

31. anonymous

i can't solve it. sorry i am really bad at it

32. anonymous

so its 59 = sqrt (122) * sqrt (41) cos a

33. phi

the contraction of "it is" is "it's" (sorry , pet peeve!)

34. phi

to find cos a, divide both sides of the equation by (sqrt(122)*sqrt(41))

35. anonymous

lol sorry i am nervous right now

36. anonymous

i get 0.83

37. phi

you can type acos(59/(sqrt(122)*sqrt(41))) in degrees in the google search window to find the angle or use a calculator.

38. anonymous

i get 0.58

39. phi

that's in radians. I would do 2 things. first, use a few more digits in your answer to 59/(sqrt(122)*sqrt(41)) (because your book is finding the answer rounded to tenths of a degree) then use degree mode with the calculator or type in degrees when using google

40. anonymous

oh k i get 33.5 degrees

41. phi

good, matches what I got.

42. anonymous

but thats just one angle. how do i find the other ones

43. phi

If you want the angle with vertex P, which we would name as angle QPR (or angle RPQ), form the vectors (q-p) and (r-p) and do the same thing: (q-p) dot (r-p) = |(q-p)| |(r-p)| cos P

44. anonymous

oh k

45. anonymous

would you help me with other question that i posted?

46. anonymous

and to find the third angle, i could add two angles and make it equal to 180 right?

47. phi

btw, notice that (q-p) = -(p-q), so just negate each element in your (p-q) vector , which you already know. And of course the length of (q-p) = length (p-q)

48. phi

Yes, the third angle = 180 - sum(other two) but it doesn't hurt to do the dot product.

49. phi

as a check, and to get comfortable with the procedure

50. anonymous

yes i need practice.

51. phi

52. anonymous

ok

53. anonymous

i get 31.8 degrees

54. anonymous

what would be the setup to find the last angle?

55. phi

yes, looks good. for the last angle, the vertex is R look at how you did the previous two, and look for the pattern. post your setup

56. anonymous

(p-r) * (q-r) = |p-r| |q-r| cos a ?

57. anonymous

i get 114.8 for the last one

58. phi

notice that 180 - (31.8+33.5)= 114.7 which matches your result within rounding error.

59. anonymous

yes

60. anonymous