Question

The vectors from the origin to the points A, B,

C and D are A= i + j + k

B = 2i +3j

C= 3i + 5j -2k

D = k- j

Show that the lines AB and CD are parallel and find the ratios of their lengths

Answer 1

Find how the lines can be described in unit vectors:

$$AB=\vec B-\vec A=(2\hat i+3\hat j)-(\hat i+\hat j+\hat k)=\\=\hat i+2\hat j-\hat k,\\\space\\ CD=\vec D-\vec C=(-\hat j+\hat k)-(3\hat i+5\hat j-2\hat k)=\\=-3\hat i-6\hat j+3\hat k.$$

Find the dot product:

$$AB·CD=(\hat i+2\hat j-\hat k)·(-3\hat i-6\hat j+3\hat k)=\\ =-3-12-3=-18.$$

Find the length of individual line:

$$|AB|=\sqrt{1^2+2^2+(-1)^2}=\sqrt 6.\\ |CD|=\sqrt{(-3)^2+(-6)^2+3^2}=\sqrt54.\\\space\\ \dfrac{|CD|}{|AB|}=3.$$

Since the dot product is

$$AB·CD=|AB|·|CD|\cos\theta,$$

we can find the angle $\theta$ between the lines to establish the configuration of the lines relative to each other:

$$\theta=\arccos\dfrac{AB·CD}{|AB|·|CD|}=180°.$$

As we see, the lines are parallel to each other.