Answer to Question #114893 in Analytic Geometry for Vishalia

Question

QUESTION 8 8.1 Let L1 and L2 be lines defined by x = w0 + su, s ∈ R and y = w1 + tv, t ∈ R, respectively. Show that L1and L2 are parallel if and only if u = kv for some k ∈ R.

Answer 1

Let's consider two lines $L_1\colon \vec{x}=\vec w_0+s \vec u$ and $L_2\colon \vec y=\vec w_1+t \vec v$.

Parallelism of lines is equivalent to proportionality of their directing vectors.

We have two directing vectors $\vec{u}$ and $\vec{v}$. Hence, $L_1 \parallel L_2 \iff \vec u=k \vec v$ for some $k \in \mathbb R$