does $\frac{d(p_1,p_2)}{d(p_2,p_3)}=\frac{d(q_1,q_2)}{d(q_2,q_3)}$

Question

the problem is :

if $p_1,p_2,p_3$ are collinier points in the space , and for an arbitrary line $l$ we have that $q_1,q_2,q_3$ are the image of $p_1,p_2,p_3$ on $l$ respectivly. then does the following hold? $$\frac{d(p_1,p_2)}{d(p_2,p_3)}=\frac{d(q_1,q_2)}{d(q_2,q_3)}$$

i only trid to take a plane $W$ such that $l \in W$ and then put image of $p_1,p_2,p_3$ on $W$. but i could not move forward.

The image of $p$ on $l$ is the point $q$ on $l$ such that $pq\perp l$? $d(p,q)$ is the euclidean distance between $p,q$? — Shubham Johri, Feb 15 '21 at 05:25

score 0 · Answer 1 · edited Feb 15 '21 at 05:42

0

Hint: Let $\hat l$ denote the unit vector along $l$ and $\theta$ be the angle between the line containing $p_i$ and $l$. Then$$d(q_1,q_2)=\left|\left[\vec{p_2}-\vec{p_1}\right]\cdot\hat l\right|=d(p_1,p_2)\left|\cos\theta\right|$$

edited Feb 15 '21 at 05:42

VIVID

11,604

answered Feb 15 '21 at 05:36

Shubham Johri

17,659

does $\frac{d(p_1,p_2)}{d(p_2,p_3)}=\frac{d(q_1,q_2)}{d(q_2,q_3)}$

1 Answers1