First slide

Straight lines in 3D

Question

$Given lines \frac{x - 4}{2} = \frac{y + 5}{4} = \frac{z - 1}{- 3} and \frac{x - 2}{1} = \frac{y + 1}{3} = \frac{z}{2}$
$Statement (S_{1}) : The lines intersect$
$Statement (S_{2}) : They are not parallel$

Moderate

A

$(S_{1}) is True, (S_{2}) is True; (S_{2}) is a correct explanation for (S_{1})$
B

$(S_{1}) is True, (S_{2}) is True; (S_{2}) is not a correct explanation for (S_{1})$
C

$(S_{1}) is True, (S_{2}) is False$
D

$(S_{1}) is False, (S_{2}) is True$

Solution

$l e t A = (4, - 5, 1), B = (2, - 1, 0) \bar{c} = (2, 4, - 3), \bar{d} = (1, 3, 2)$
$t h e n [\bar{A B} \bar{c} \bar{d}] = |\begin{matrix} - 2 & 4 & - 1 \\ 2 & 4 & - 3 \\ 1 & 3 & 2 \end{matrix}| \neq 0$
$∴ T h e l i n e s a r e n o n - c o p l a n a r$
$\bar{c} i s n o t p a r a l l e l t o \bar{d} \Rightarrow T h e l i n e s a r e n o n - p a r a l l e l$
$H e n c e t h e y a r e s k e w - l i n e s$

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