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 September 13th, 2012, 09:06 PM #1 Newbie   Joined: Sep 2012 Posts: 10 Thanks: 0 Not sure if this belongs here but i'll give it a shot Hi, i'm not sure if this is the correct place to post this but im having trouble with this question Find two vectors v?1 and v?2 whose sum is ?0,2,5?, where v?1 is parallel to ?1,0,?4? while v?2 is perpendicular to ?1,0,?4? I was able to find that y1=0 and y2=2 but am having trouble finding out what x1, x2, z1, z2 are Any help would be great Thanks
September 13th, 2012, 10:02 PM   #2
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Re: Not sure if this belongs here but i'll give it a shot

Quote:
 $\text{Find two vectors }\vec{v_1}\text{ and }\vec{v_2}\text{ whose sum is }\langle0,\,2,\,5\rangle, \;\;\;\text{where }\vec{v_1}\,\parallel\,\langle1,\,0,\,-4\rangle\,\text{ and }\,\vec{v_2}\,\perp \,\langle1,\,0,\,-4\rangle$

$\text{Let: }\:\begin{Bmatrix}\vec{v_1}=&\langle x_1,\,y_1,\,z_1\rangle \\ \\ \\ \vec{v_2}=&\langle x_2,\,y_2,\,z_2\rangle \end{Bmatrix}=$

$\text{Their sum is }\langle 0,2,5\rangle:\;\;\begin{Bmatrix}x_1+x_2 \:=\:0 & \;\;\Rightarrow\;\; & x_2 &=& -x_1 \\ \\ \\
y_1+y_2 \:=\:2 & \Rightarrow & y_2 &=& 2-y_1 \\ \\ \\
z_1+z_2\:=\:5 & \Rightarrow & z_2 &=& 5-z_1 \end{Bmatrix}$
[color=beige] .[/color][color=blue][1][/color]

$\vec{v_1}\,\parallel\,\langle1,0,-4\rangle \;\;\;\Rightarrow\;\;\; \langle x_1,y_1,z_1\rangle \,=\,a\langle 1,0,-4\rangle \;\;\;\Rightarrow\;\;\; \begin{Bmatrix}x_1 &a \\ \\ y_1=&0 \\ \\ z_1=&-4a \end{Bmatrix}=$[color=beige] .[/color][color=blue][2][/color]

$\vec{v_2}\,\perp\,\langle 1,0,-4\rangle \;\;\;\Rightarrow\;\;\;\langle x_2,y_2,z_2\rangle\cdot\langle1,0,-4\rangle \:=\:0 \;\;\;\Rightarrow\;\;\; x_2\,-\,4z_2 \:=\:0$

Substitute [color=blue][1][/color]:[color=beige] .[/color]$-x_1\,-\,4(5\,-\,z_1) \:=\:0 \;\;\;-x_1\,-\,20\,+\,4z_1\:=\:0$

Substitute [color=blue][2][/color]:[color=beige] .[/color]$-a\,-\,20\,+\,4(-4a) \:=\:0 \;\;\;-17a \:=\:20 \;\;\;\Rightarrow\;\;\;a \:=\:-\frac{20}{17}$

Substitute into [color=blue][2][/color]:[color=beige] .[/color]$\begin{Bmatrix}x_1=&-\frac{20}{17} \\ \\ y_1=&0 \\ \\ z_1=&\frac{80}{17\end{Bmatrix} \;\;\;\Rightarrow\;\;\;\vec{v_1} \;=\;\left\langle -\frac{20}{17},\:0,\:\frac{80}{17}\right\rangle$

Substitute into [color=blue][1][/color]:[color=beige] .[/color]$\begin{Bmatrix}x_2=&\frac{20}{17} \\ \\ y_2=&2 \\ \\ z_2=&\frac{5}{17} \end{Bmatrix} \;\;\;\Rightarrow\;\;\;l\vec{v_2} \;=\;\left\langle \frac{20}{17},\:2,\:\frac{5}{17}\right\rangle$

 September 13th, 2012, 10:30 PM #3 Newbie   Joined: Sep 2012 Posts: 10 Thanks: 0 Re: Not sure if this belongs here but i'll give it a shot Thanks a lot soroban

 Tags belongs, give, shot

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# The addition of two vectors and gives a third vector known as the --------------of the two vectors

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