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Math Help - Equation of a line in 2-space

  1. #1
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    Equation of a line in 2-space

    I have two lines

    r: [x,y]=[3,2]+t[4,-5]
    q: [x,y]=[1,1]+s[7,k]

    a) for what values of k are the lines perpendicular?

    I got this by using the dot product of [4,-5].[7,k]=0 and then solving for k.

    b) for what value of k are the lines paralell?

    I'm not sure how to do this. I know the the normal vector is n=[A,B] and the direction vector is m=[B,-A].

    so I took the dot product of [5,4].[7,k] =1
    35+4k=1
    k=-(34/4)

    I made the dot product equal to 1 since cos 0 is 1.
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  2. #2
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    Quote Originally Posted by darksoulzero View Post
    I have two lines

    r: [x,y]=[3,2]+t[4,-5]
    q: [x,y]=[1,1]+s[7,k]

    ...
    b) for what value of k are the lines paralell?

    I'm not sure how to do this. I know the the normal vector is n=[A,B] and the direction vector is m=[B,-A].

    so I took the dot product of [5,4].[7,k] =1
    35+4k=1
    k=-(34/4)

    I made the dot product equal to 1 since cos 0 is 1.
    Two lines are parallel (in 2-D) if the normal vectors are collinear, that means if

    \overrightarrow{n_1}=c\cdot \overrightarrow{n_2}

    where c is a real constant.

    Plug in the known vectors into the equation above. Solve for c and then for k. You should come out with k = -\frac{35}4
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    Re:

    hi,
    i have something interesting for you about the query you have asked.
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    Quote Originally Posted by alinora11 View Post
    hi,
    i have something interesting for you about the query you have asked.
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  5. #5
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    Quote Originally Posted by darksoulzero View Post
    I have two lines

    r: [x,y]=[3,2]+t[4,-5]
    q: [x,y]=[1,1]+s[7,k]

    a) for what values of k are the lines perpendicular?

    I got this by using the dot product of [4,-5].[7,k]=0 and then solving for k.

    b) for what value of k are the lines paralell?

    I'm not sure how to do this. I know the the normal vector is n=[A,B] and the direction vector is m=[B,-A].

    so I took the dot product of [5,4].[7,k] =1
    35+4k=1
    k=-(34/4)

    I made the dot product equal to 1 since cos 0 is 1.
    The dot product of two vectors, a and b, is |a||b|cos(\theta). Of course, if the lines are parallel, cos(\theta)= 1. If you really want to use the dot product, you would want it equal to \sqrt{25+ 16}\sqrt{49+ k}, not 1.

    But much simpler is that if two vectors are parallel, one is a multiple of the other. You must have [5,4]= m[7, k] so you have two equations, 5= 7m and 4= mk to solve for the two unknowns. Since you really only want k, write m= 5/7, from the first equation, and replace m by 5/7 in the second equation.
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