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

• Feb 17th 2009, 04:43 PM
narutoblaze
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:D Will be learning more hopefully haha:D
Thanks

• Feb 17th 2009, 05:48 PM
Reckoner
Quote:

Originally Posted by narutoblaze
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:

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

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

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

Quote:

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

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

Quote:

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

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

Quote:

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

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

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

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

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

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

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

$\Rightarrow-2L=4+9+16=29\Rightarrow L=-\frac{29}2$
• Feb 17th 2009, 06:29 PM
narutoblaze
That looks uber complicating but you have the correct answer thanks!(Bow)
I'm gonna show this to my teacher for further clearification but I bet she won't know rofl.