First slide

Straight lines in 3D

Question

If the lines $\frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 1}{4}$ and $\frac{x - 3}{1} = \frac{y - k}{2} = \frac{z}{1}$ intersect then k =

Moderate

A

-1
B

2/9
C

9/2
D

0

Solution

Lines are intersecting
$|\begin{matrix} x_{2} - x_{1} & y_{2} - y_{1} & z_{2} - z_{1} \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{matrix}| = 0$
$|\begin{matrix} - 2 & - 1 - k & 1 \\ 2 & 3 & 4 \\ 1 & 2 & 1 \end{matrix}| = 0 \Rightarrow k = \frac{9}{2}$

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