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Thread: "Any linear combination of these vectors will be perpendicular to (a,b,c)."

  1. #1
    s3a
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    "Any linear combination of these vectors will be perpendicular to (a,b,c)."

    For Question 4, here ( https://www.docdroid.net/LubIxXM/e1.pdf - scroll down for the answer for Question 4, to which I am referring ), the answer says that "any linear combination of these vectors will be perpendicular to (a,b,c)."

    I confirmed one such case, here ( http://www.wolframalpha.com/input/?i=(-b... ), however, I don't understand why this is the case.

    Could someone please explain to me why that is always the case?

    Any input would be GREATLY appreciated!
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    Re: "Any linear combination of these vectors will be perpendicular to (a,b,c)."

    Quote Originally Posted by s3a View Post
    For Question 4, here ( https://www.docdroid.net/LubIxXM/e1.pdf - scroll down for the answer for Question 4, to which I am referring ), the answer says that "any linear combination of these vectors will be perpendicular to (a,b,c)."
    Here are three: $<0,-c ,b >,~<c, 0, -a>,~\&~<-b,a , 0>$
    To see that, what is the dot product of each of those with $<a,b,c>~?$
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    s3a
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    Re: "Any linear combination of these vectors will be perpendicular to (a,b,c)."

    Thanks for your response.

    Having said that, I see that their dot products are 0, and I know that that means that they are perpendicular. My problem is with the "linear combination" part. Could you please elaborate on that part? More specifically, I know how to find linear combinations of vectors, but I don't understand why the linear combination of a set of vectors that are each perpendicular to a certain vector (vector (a,b,c), in this case) yields another vector that is also perpendicular to that certain vector (vector (a,b,c), in this case).
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    Re: "Any linear combination of these vectors will be perpendicular to (a,b,c)."

    Quote Originally Posted by s3a View Post
    Thanks for your response.

    Having said that, I see that their dot products are 0, and I know that that means that they are perpendicular. My problem is with the "linear combination" part. Could you please elaborate on that part? More specifically, I know how to find linear combinations of vectors, but I don't understand why the linear combination of a set of vectors that are each perpendicular to a certain vector (vector (a,b,c), in this case) yields another vector that is also perpendicular to that certain vector (vector (a,b,c), in this case).
    Well that is a general theorem.
    Suppose that each of $\vec{a}~\&~\vec{b}$ is perpendicular to $\vec{v}$.
    Consider the lin. com, $\alpha\vec{a}+\beta\vec{b}$ then
    $(\alpha \vec{a}+\beta\vec{b})\cdot\vec{v}\\=(\alpha \vec{a}\cdot\vec{v}+\beta\vec{b}\cdot\vec{v})\\= ( \alpha
    \cdot 0 +\beta\cdot 0)\\=0$
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    s3a
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    Re: "Any linear combination of these vectors will be perpendicular to (a,b,c)."

    That was what I wanted to know! Thank you very much!
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