1. ## Urgent!! triple product

Show that the lines

x=a1s+ b1, y= a2s + b2 , z =a3s+b3 , -infinity<s<infinity

and

x=c1t+ d1, y= c2t + d2 , z =c3t+d3 , -infinity<t<infinity

intersect or parallel if and only if

| a1 c1 b1-d1 |
| a2 c2 b2-d2 | = 0
| a3 c3 b3-d3 |

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3. Hello, fatduck88!

I have a start on this problem anyway . . .

Show that the lines: . $\begin{Bmatrix}x&=&a_1s+ b_1 \\y&=& a_2s + b_2 \\ z &=&a_3s+b_3\end{Bmatrix}\;\text{ and }\;\begin{Bmatrix}x&=&c_1t+ d_1 \\ y&=&c_2t + d_2 \\ z&=&c_3t+d_3 \end{Bmatrix}$

. . intersect or are parallel if and only if: . $\begin{vmatrix}\:a_1 & c_1 &(b_1\!-\!d_1)\: \\ a_2 & c_2 & (b_2\!-\!d_2) \\ a_3 & c_3 & (b_3\!-\!d_3) \end{vmatrix} \;=\;0$

The lines intersect if: . $\begin{array}{ccc}a_1s+b_1 &=& c_1t+d_1 \\ a_2s+b_2 &=& c_2t+d_2 \\ a_3s+b_3 &=&c_3t+d_3 \end{array}$

. . That is, if the system: . $\begin{array}{ccc}a_1s - c_1t + (b_1-d_1) &=&0 \\ a_2s - c_2t + (b_2-d_2) &=& 0 \\ a_3s - c_3t + (b_3-d_3) &=&0 \end{array}$ . has a solution $(s,t)$

Now we must relate this system to that determinant . . .

[But I truly dislike "if and only if" problems . . . p'too!]
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