L1 :X-4=Y-8=Z+1L2 :X-16=Y-2=Z+1

2 3 -4 -6 1 2

A) show that L1 and L2 intersected

B) find parametric equation of the line that passes through the point of intersection of L1 and L2 IS perpandicular to both of them

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- Apr 12th 2008, 11:01 AMfastman390symmetric equation
L1 :

__X-4__=__Y-8__=__Z+1__L2 :__X-16__=__Y-2__=__Z+1__

2 3 -4 -6 1 2

A) show that L1 and L2 intersected

B) find parametric equation of the line that passes through the point of intersection of L1 and L2 IS perpandicular to both of them - Apr 12th 2008, 11:59 AMMathstud28Ok
- Apr 12th 2008, 12:10 PMearboth
Re-write the equation:

$\displaystyle L_1:\left\{\begin{array}{lcr}x&=&2t+4 \\ y&=&3t+8 \\ z&=&-4t-1\end{array}\right.$ ..... and ..... $\displaystyle L_2:\left\{\begin{array}{lcr}x&=&-6s+16 \\ y&=&s+2 \\ z&=&2s-1\end{array}\right.$

to #A) Solve the system of simultaneous equations:

$\displaystyle \left|\begin{array}{lcr}-6s+16&=&2t+4 \\ s+2&=&3t+8 \\ 2s-1&=&-4t-1\end{array}\right.$ ..... which will yield $\displaystyle s=\frac{12}5~\wedge~t=-\frac65$

The point of intersection is $\displaystyle P\left(\frac85\ ,\ \frac{22}5\ ,\ \frac{19}5\right)$

to #B: The direction vector of the line L3 is the crossproduct of the direction vectors of L1 and L2:

$\displaystyle (2, 3, -4) \times (-6, 1, 2) = (10, 20, 20)$

and therefore the parametric equation of L3 is:

$\displaystyle L_3:\left\{\begin{array}{lcr}x&=&10r+\frac85 \\ \\ y&=&20r+\frac{22}5 \\ \\ z&=&20r+\frac{19}5\end{array}\right.$