Proof of Prop: Does a=b for all x in R3?

[nicolauslamsiu]

Prop: Suppose a and b be vectors in R3. If a·x=b·x for all vector x in R3, then a=b

My question if the proposition is always true.
And if x is a zero vector, is the proposition still valid?

 

[andrewkirk]

And if x is a zero vector, is the proposition still valid?

Note the words 'for all vectors x in R3', in the statement. Not 'for some vector x in R3'.

To see whether it's true for all pairs a,b, consider what you can conclude if it's true for all three of the basis vectors (1 0 0), (0 1 0), (0 0 1).

 

[nicolauslamsiu]

Aha, i have missed something. Thanks for ur help@@

 

[jbunniii]

Another proof: $a \cdot x = b \cdot x$ if and only if $(a-b) \cdot x = 0$. If this holds for all $x$, then choosing $x = a-b$ implies that $(a-b)\cdot (a-b) = 0$. Since $(a-b)\cdot (a-b)$ is the square of the length of $a-b$, this means that $a-b$ has length zero, so $a-b = 0$, and therefore $a=b$.