Question

Determine whether the following lines represented by the vector equations below intersect, are parallel, are skew, or are identical.
r(t)=⟨1−t,3+2t,−3t⟩
s(t)=⟨2t,−3−4t,3+6t⟩

Answer 1

Answer:

r(t) and s(t) are parallel.

Step-by-step explanation:

Given that :

the  lines represented by the vector equations are:

r(t)=⟨1−t,3+2t,−3t⟩

s(t)=⟨2t,−3−4t,3+6t⟩

The objective is to determine if the following lines represented by the vector equations below intersect, are parallel, are skew, or are identical.

NOTE:

Two lines will be parallel if $\dfrac{x_1}{x_2}= \dfrac{y_1}{y_2}= \dfrac{z_1}{z_2}$

here;

$d_1 = (-1, \ 2, \ -3)$

Thus;

$r(t) = \dfrac{x-1}{-1} = \dfrac{y-3}{2}=\dfrac{z-0}{-3} = t$

$d_2 =(2, \ -4, \ +6)$

$s(t) = \dfrac{x-0}{2} = \dfrac{y+5}{-4}=\dfrac{z-3}{6} = t$

∴

$\dfrac{d_1}{d_2}= \dfrac{-1}{2} = \dfrac{2}{-4}= \dfrac{-3}{-6}$

Hence, we can conclude that r(t) and s(t) are parallel.