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Thread: Gr 12. Vector Question Involving Dot Product of Two Vectors(Part 3)

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    Gr 12. Vector Question Involving Dot Product of Two Vectors(Part 3)

    Okay, the last one for today hopefully phew...!
    I can guaranteed this one is DIFFICULT!!!

    Three vectors x, y, and z satisfy x + y + z = 0. Calculate the value of $\displaystyle \vec x \cdot \vec y\ + \vec y \cdot \vec z\ + \vec z \cdot \vec x$, is |x| = 2, |y| = 3, and |z| = 4.

    Lol learning how to use LateX here to see if it helps Will be learning more hopefully haha
    Thanks

    Last edited by narutoblaze; Feb 17th 2009 at 04:17 PM.
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    Quote Originally Posted by narutoblaze View Post
    Okay, the last one for today hopefully phew...!
    I can guaranteed this one is DIFFICULT!!!
    Okay, let's say the vectors have the following components:

    $\displaystyle \textbf x = x_1\textbf i+x_2\textbf j+x_3\textbf k$

    $\displaystyle \textbf y = y_1\textbf i+y_2\textbf j+y_3\textbf k$

    $\displaystyle \textbf z = z_1\textbf i+z_2\textbf j+z_3\textbf k$

    Three vectors x, y, and z satisfy x + y + z = 0.
    This tells us that

    $\displaystyle \left\{\begin{array}{c}
    x_1+y_1+z_1=0\\
    x_2+y_2+z_2=0\\
    x_3+y_3+z_3=0
    \end{array}\right.$

    |x| = 2, |y| = 3, and |z| = 4
    Using the dot product property $\displaystyle \textbf u\cdot\textbf u=\lVert\textbf u\rVert^2,$ we have

    $\displaystyle \left\{\begin{array}{rcl}
    x_1^2+x_2^2+x_3^2 & = & 4\\
    y_1^2+y_2^2+y_3^2 & = & 9\\
    z_1^2+z_2^2+z_3^2 & = & 16
    \end{array}\right.$

    Calculate the value of $\displaystyle \vec x \cdot \vec y\ + \vec y \cdot \vec z\ + \vec z \cdot \vec x$
    In terms of the components, what we are trying to find is

    $\displaystyle (x_1y_1+x_2y_2+x_3y_3)+(y_1z_1+y_2z_2+y_3z_3)+(z_1 x_1+z_2x_2+z_3x_3)=L$

    Rearranging and substituting from the above, we have

    $\displaystyle \begin{array}{rcl}
    2L&=&(x_1y_1+x_2y_2+x_3y_3)+(z_1x_1+z_2x_2+z_3x_3) \\
    &+&(y_1z_1+y_2z_2+y_3z_3)+(z_1x_1+z_2x_2+z_3x_3 )\\
    &+&(x_1y_1+x_2y_2+x_3y_3)+(y_1z_1+y_2z_2+y_3z_3 )
    \end{array}$

    $\displaystyle \Rightarrow\begin{array}{rcl}
    2L&=&[(x_1y_1+x_2y_2+x_3y_3)+(z_1x_1+z_2x_2+z_3x_3)]\\
    &+&[(y_1z_1+y_2z_2+y_3z_3)+(z_1x_1+z_2x_2+z_3x_3)]\\
    &+&[(x_1y_1+x_2y_2+x_3y_3)+(y_1z_1+y_2z_2+y_3z_3)]
    \end{array}$

    $\displaystyle \Rightarrow\begin{array}{rcl}
    2L&=&[x_1y_1+z_1x_1+x_2y_2+z_2x_2+x_3y_3+z_3x_3]\\
    &+&[y_1z_1+z_1x_1+y_2z_2+z_2x_2+y_3z_3+z_3x_3]\\
    &+&[x_1y_1+y_1z_1+x_2y_2+y_2z_2+x_3y_3+y_3z_3]
    \end{array}$

    $\displaystyle \Rightarrow\begin{array}{rcl}
    2L&=&[x_1(y_1+z_1)+x_2(y_2+z_2)+x_3(y_3+z_3)]\\
    &+&[z_1(x_1+y_1)+z_2(x_2+y_2)+z_3(x_3+y_3)]\\
    &+&[y_1(z_1+x_1)+y_2(z_2+x_2)+y_3(z_3+x_3)]
    \end{array}$

    $\displaystyle \Rightarrow\begin{array}{rcl}
    2L&=&[x_1(-x_1)+x_2(-x_2)+x_3(-x_3)]\\
    &+&[z_1(-z_1)+z_2(-z_2)+z_3(-z_3)]\\
    &+&[y_1(-y_1)+y_2(-y_2)+y_3(-y_3)]
    \end{array}$

    $\displaystyle \Rightarrow-2L=\left(x_1^2+x_2^2+x_3^2\right)+\left(z_1^2+z_2^ 2+z_3^2\right)+\left(y_1^2+y_2^2+y_3^2\right)$

    $\displaystyle \Rightarrow-2L=4+9+16=29\Rightarrow L=-\frac{29}2$
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  3. #3
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    That looks uber complicating but you have the correct answer thanks!
    I'm gonna show this to my teacher for further clearification but I bet she won't know rofl.
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