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anonymous

  • one year ago

Find the angle between the given vectors to the nearest tenth of a degree. u = <6, -1>, v = <7, -4>

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  1. freckles
    • one year ago
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    have you tried using the formula for finding the angle between two vectors?

  2. freckles
    • one year ago
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    solve for theta \[\cos(\theta)=\frac{ u \cdot v}{|u||v|}\]

  3. anonymous
    • one year ago
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    cos(theta) = 6 * 7 / |-1| |-4| ?

  4. freckles
    • one year ago
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    how did you get that?

  5. anonymous
    • one year ago
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    i just plug in the numbers from the question to the formula you gave me. im sorry, i really dont know how to do this..

  6. freckles
    • one year ago
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    it doesn't look like you tried to find the dot product of the two vectors nor the magnitude of each vector dot product: you multiply corresponding components and then add those results: example: \[<x_1,x_2> \cdot <y_1,y_2>=x_1 y_1 +x_2 y_2\] to find the magnitude, square each component then add those results and take the square root of that sum example: \[|<x_1,x_2>|=\sqrt{x^2_1+x^2_2}\]

  7. anonymous
    • one year ago
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    dot product: 42 + 4 = 46 magnitude: 36 + 1 = 37 = 6.08 ?

  8. freckles
    • one year ago
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    \[u \cdot v=<6,-1> \cdot <7,-4>=6(7)+(-1)(-4)=42+4=46 \\ |u|=\sqrt{6^2+(-1)^2}=\sqrt{36+1}=\sqrt{37}\] almost you forgot to take the square root

  9. freckles
    • one year ago
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    you still need to find |v|

  10. anonymous
    • one year ago
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    i took the square root of 36 + 1 and got 6.08 do i find |v| then same way?

  11. freckles
    • one year ago
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    well if you mean by what I told you above yes \[|<x_1,x_2>|=\sqrt{x_1^2+x^2_2}\] this is not going to change because you have different numbers in your vector...

  12. anonymous
    • one year ago
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    49 + 16 = 65 square root = 8.06

  13. freckles
    • one year ago
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    sqrt(49+16)=sqrt(65) then this will be correct

  14. freckles
    • one year ago
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    \[u \cdot v=46 \\ |u|=\sqrt{37} \\ |v|=\sqrt{65}\]

  15. freckles
    • one year ago
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    enter this into the formula I gave above and solve for theta

  16. anonymous
    • one year ago
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    okay, so the formula is cos(theta) = u * v / |u| |v| cos(theta) - u * v / |46| |√37| what would be the values of u and v? <x1,y1> ?

  17. freckles
    • one year ago
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    you do realize the formula is: \[\cos(\theta)=\frac{ u \cdot v}{|u||v|}\] we already found u dot v in the numerator and we already found ||u| and |v| in denominator

  18. freckles
    • one year ago
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    I even summarized these values above

  19. freckles
    • one year ago
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    \[u \cdot v=46 \\ |u|=\sqrt{37} \\ |v|=\sqrt{65}\] \[\cos(\theta)=\frac{ u \cdot v}{|u||v|}\] \[\cos(\theta)=\frac{46}{\sqrt{37} \sqrt{65}}\]

  20. anonymous
    • one year ago
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    cos(theta) = 46 / √37 √65

  21. anonymous
    • one year ago
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    oh, okay.. i see

  22. freckles
    • one year ago
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    last step solve for theta theta is the angle after all

  23. anonymous
    • one year ago
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    i got 60.96..

  24. freckles
    • one year ago
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    maybe you entered something wrong I'm getting this: http://www.wolframalpha.com/input/?i=arccos%2846%2F%28sqrt%2837%29*sqrt%2865%29%29%29

  25. anonymous
    • one year ago
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    i see, so it would be 20.3 degrees

  26. freckles
    • one year ago
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    yes but you need to figure out what you are doing in your calculator so you can get that answer by yourself

  27. anonymous
    • one year ago
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    yeah i know, i found out what i did wrong.

  28. freckles
    • one year ago
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    well that's great

  29. anonymous
    • one year ago
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    thank you so much for your help! you are awesome!

  30. freckles
    • one year ago
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    it was no problem

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