Results 1 to 3 of 3

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

  1. #1
    Junior Member
    Joined
    Feb 2009
    Posts
    27

    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 \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; February 17th 2009 at 04:17 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Reckoner's Avatar
    Joined
    May 2008
    From
    Baltimore, MD (USA)
    Posts
    1,024
    Thanks
    75
    Awards
    1

    Smile

    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:

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

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

    \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

    \left\{\begin{array}{c}<br />
x_1+y_1+z_1=0\\<br />
x_2+y_2+z_2=0\\<br />
x_3+y_3+z_3=0<br />
\end{array}\right.

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

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

    Calculate the value of \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

    (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

    \begin{array}{rcl}<br />
2L&=&(x_1y_1+x_2y_2+x_3y_3)+(z_1x_1+z_2x_2+z_3x_3)  \\<br />
&+&(y_1z_1+y_2z_2+y_3z_3)+(z_1x_1+z_2x_2+z_3x_3  )\\<br />
&+&(x_1y_1+x_2y_2+x_3y_3)+(y_1z_1+y_2z_2+y_3z_3  )<br />
\end{array}

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

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

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

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

    \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)

    \Rightarrow-2L=4+9+16=29\Rightarrow L=-\frac{29}2
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Feb 2009
    Posts
    27
    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.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Vector part of calculus course question
    Posted in the Calculus Forum
    Replies: 2
    Last Post: January 10th 2012, 07:10 AM
  2. proove involving dot product and zero vector
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: February 10th 2010, 07:16 AM
  3. Replies: 4
    Last Post: February 17th 2009, 03:33 PM
  4. Replies: 2
    Last Post: February 17th 2009, 03:27 PM
  5. Question involving vectors
    Posted in the Calculus Forum
    Replies: 0
    Last Post: February 5th 2008, 04:12 AM

Search Tags


/mathhelpforum @mathhelpforum