Should I prove the opposite direction?

Question

Let $A$, $B$, $O$ be distinct points in a plane. Show that $P$ is on the line through $A$ and $B$ if and only if $\vec{OP}$ can be written in the form $\vec{OP} = (1-t)\vec{OA} + (t)\vec{OB}$

When we say that $y=ax+b$, we can see the points of $\mathbb R^2$ that satisfy the equation. One normally does not think both ways

1. the set of points with some property satisfy the equation,
2. the points described by the equation has the property.

If the above vector equation begs the converse proof, why is it so? Why do I get the feeling that its construction is enough to prove both the "if and only if" directions?

Answer 1

You are essentially asking:

1. isn't it obvious that $y=ax+b$ describes the line with $y$-intercept $b$ and gradient $a$ ?
2. doesn't the equation $\vec{OP} = (1-t)\vec{OA} + (t)\vec{OB}$ intrinsically describe the line $AB$ ?

The answer is No: the implication "every point satisfying the equation ${\vec{OP} = (1-t)\vec{OA} + (t)\vec{OB}}$ lies on the line $AB$"—that is, the geometric property of that vector equation—while indeed "obvious", is true by derivation rather than by definition or axiom.