Is $p-t(p+q)$ the same as $p+t(q-p)$

Question

This might be a silly question, but is $p-t(p+q)$ the same as $p+t(q-p)$?

Not that it matters, but this is a formula in linear algebra called two-point form. The original formula is $(1-q)p+tq$ and I am attempting to simplify it.

Answer 1

is $p-t(p+q)$ the same as $p+t(q-p)$?

The first one is $p-tp-tq$ and the second one is $p+tq-tp$.

The original formula is $(1-q)p+tq$ and I am attempting to simplify it.

There is not really any simplification to be done, although you can rearrange things. For example: $(1-t)p+tq=p-tp+tq=p-t(p-q)=p+t(q-p)$

The original form is already pretty useful because one can see why it defines a segment between $p$ and $q$ (assuming you are using something like vectors.) When $t=0$ you can see you are at the point $p$, and when $t=1$ you have arrived at $q$, and everything else in $(0,1)$ puts you somewhere between.

Answer 2

$$(1-t)p + tq = p - tp + tq = p + t(q-p) .$$

Answer 3

Just multiply each out:

$$p-t(p+q)=p - tp - tq$$

$$p+t(q-p)=p - tp + tq$$

Not the same!