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Thread: find 2 unit-vectors in 2-space that makes an angle of 45 degrees with 4i+3j.

  1. #1
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    find 2 unit-vectors in 2-space that makes an angle of 45 degrees with 4i+3j.

    Hi .


    we know that u and v are unit vectors , therefore, =1.

    norm of u gets: x^2+y^2=1
    norm of v gets: = 5.

    u*v= norm of u*norm of v(cos (Pi/4))
    (xi+yj)*(4i+3y)=5(1/radical (2))
    x=5/4radical(2)-3/4y

    then I just solve for x, from norm of u, and solve for y then. then I will have answer with 2 unit vectors. I am wondering if there is any easier way to do this problem because the algebra is a bit intensive, ty.
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    Re: find 2 unit-vectors in 2-space that makes an angle of 45 degrees with 4i+3j.

    Quote Originally Posted by math951 View Post
    we know that u and v are unit vectors , therefore, =1.

    norm of u gets: x^2+y^2=1
    norm of v gets: = 5.

    u*v= norm of u*norm of v(cos (Pi/4)) WHAT?
    (xi+yj)*(4i+3y)=5(1/radical (2))
    x=5/4radical(2)-3/4y
    I think that this is the most confused post that we have seen in the last five years.
    Please just post a well written question as given. Do not try to put your understanding on the question.
    Just post the actual wording of the problem. Then post what you have done to solve the problem.
    Please do that.
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  3. #3
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    Re: find 2 unit-vectors in 2-space that makes an angle of 45 degrees with 4i+3j.

    Hey math951.

    Hint - Try taking the vector and put in polar form and add that angle and see what you get.
    Thanks from math951
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    Re: find 2 unit-vectors in 2-space that makes an angle of 45 degrees with 4i+3j.

    Quote Originally Posted by math951 View Post
    Hi .


    we know that u and v are unit vectors , therefore, =1.
    You mean "have norm 1", not "are equal to 1".

    norm of u gets: x^2+y^2=1
    norm of v gets: = 5.
    You just said that a unit vector has norm 1! The norms of both u and v are 1.

    u*v= norm of u*norm of v(cos (Pi/4))
    v is NOT 4i+ 3y! "u" and "v" are two unit vectors that make angle pi/4 with 4i+ 3j. They actually form a right angle with each other. Do you see why?

    (xi+yj)*(4i+3y)=5(1/radical (2))
    x=5/4radical(2)-3/4y

    then I just solve for x, from norm of u, and solve for y then. then I will have answer with 2 unit vectors. I am wondering if there is any easier way to do this problem because the algebra is a bit intensive, ty.
    You have two unknown vectors, u and v. Write u= xi+ yj and v= ai+ bj. Then you must have x^2+ y^2= 1, a^2+ b^2= 1, 4x+ 3y= 5sqt(2)/2, and 4a+ 3b= 5sqrt(2)/2. Four equations to solve for x, y, a, and b- well, actually two equations to solve for x and y which will also give you a and b.
    Thanks from math951
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    Re: find 2 unit-vectors in 2-space that makes an angle of 45 degrees with 4i+3j.

    I think Chiro's suggestion is the easiest way to do this. Just draw a picture as in the following:

    Thanks from math951
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    Re: find 2 unit-vectors in 2-space that makes an angle of 45 degrees with 4i+3j.

    TY for advice and solutions everyone. I apologize for the post, and I am unable to edit right now just way too busy :/. Physics electromagnetism and Thermodynamics is taking up all of my time.
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    Re: find 2 unit-vectors in 2-space that makes an angle of 45 degrees with 4i+3j.

    Is their solution wrong? Calculus (9781118137925), Pg. 793, Ex. 14 :: Homework Help and Answers :: Slader

    Why are the two unit vectors denoted in their solution as : <sin, sin> <cos,cos>

    Should not a unit vector be : <cos, sin> <cos, sin>

    Ty.
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    Re: find 2 unit-vectors in 2-space that makes an angle of 45 degrees with 4i+3j.

    Look again at their solution. Let $\theta=\text{atan}(3/4)$ and $\phi=\theta+\pi/4$. Then one of the posted solutions is $(\cos(\phi),\sin(\phi))$ and similarly for the other.

    I think my previous solution is better in that the result is exact, not an approximation.
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