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Math Help - solve perpendicular vectors problem

  1. #1
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    Exclamation solve perpendicular vectors problem

    (the vectors in the problem are really in column form, but i don't know how to type that so i put them in component form)

    vectorP = 2i +5j
    vectorQ = -i+ 2j

    ☆find the value of p·q
    (I know the dot product is 8, that was ez) but the next part i don't get...

    ☆ if... s = kp - (p·q)q
    find the value of the constant k such that s is perpendicular to q

    plz help
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  2. #2
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    The dot product \mathbf{p}\cdot\mathbf{q} increases when \mathbf{p} and \mathbf{q} point in the same direction, and drops to 0 exactly when \mathbf{p} and \mathbf{q} are perpendicular.

    Therefore, \mathbf{s} will be perpendicular to \mathbf{q} when

    \begin{aligned}<br />
\mathbf{s}\cdot\mathbf{q}&=(k\mathbf{p}-(\mathbf{p}\cdot\mathbf{q})\mathbf{q})\cdot\mathbf  {q}\\<br />
&=k(\mathbf{p}\cdot\mathbf{q})-(\mathbf{p}\cdot\mathbf{q})|\mathbf{q}|^2\\<br />
&=(\mathbf{p}\cdot\mathbf{q})(k-|\mathbf{q}|^2)\\<br />
&=0.<br />
\end{aligned}

    Here, we have used the formula \mathbf{q}\cdot\mathbf{q}=|\mathbf{q}|^2.
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  3. #3
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    Quote Originally Posted by Scott H View Post
    The dot product \mathbf{p}\cdot\mathbf{q} increases when \mathbf{p} and \mathbf{q} point in the same direction, and drops to 0 exactly when \mathbf{p} and \mathbf{q} are perpendicular.

    Therefore, \mathbf{s} will be perpendicular to \mathbf{q} when

    \begin{aligned}<br />
\mathbf{s}\cdot\mathbf{q}&=(k\mathbf{p}-(\mathbf{p}\cdot\mathbf{q})\mathbf{q})\cdot\mathbf  {q}\\<br />
&=k(\mathbf{p}\cdot\mathbf{q})-(\mathbf{p}\cdot\mathbf{q})|\mathbf{q}|^2\\<br />
&=(\mathbf{p}\cdot\mathbf{q})(k-|\mathbf{q}|^2)\\<br />
&=0.<br />
\end{aligned}

    Here, we have used the formula \mathbf{q}\cdot\mathbf{q}=|\mathbf{q}|^2.
    ummm... sorry i feel kinda lame, but im in high school and i kinda got lost in the first step, plz explain, im not so good with the math...i still don't see how to get the value of k
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  4. #4
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    Given that

    8(k-|\mathbf{q}|^2)=0,

    we may divide both sides by 8, giving

    k-|\mathbf{q}|^2=0.

    Now, we may add |\mathbf{q}|^2 to both sides, giving

    k=|\mathbf{q}|^2.

    Hope this helps!
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  5. #5
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    i get it now, grazie!
    Last edited by 3k1yp2; December 7th 2009 at 06:26 PM. Reason: now i got it
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