For $x, y, z \in R^n$, show if $x \cdot y = z \cdot y$ , $x = z$

Question

Is it right for me to do the following?

$x \cdot y = z \cdot y$

$x \cdot y - z \cdot y =0$

$(x-z) \cdot y= 0$

Let y = (x-z)

$(x-z) \cdot (x-z) = 0$

x = z

Similarly for another question, For $x, y, z$ in $R^3,$ show if $ x \times y = z \times y$ , $x = z$, how do I begin proving this question

Answer 1

If the question is:

If $x,z\in\Bbb R^n$ are such that $(\forall y\in\Bbb R^n):x.y=z.y$, then $x=z$.

then your answer is correct.

In the case of the cross-product, it's a little different. Assuming this time that the question is:

If $x,z\in\Bbb R^n$ are such that $(\forall y\in\Bbb R^n):x\times y=z\times y$, then $x=z$.

this can be done as follows: if $x\ne z$ take a vector $y\ne0$ which is orthogonal to $x-y$. Then $\|(x-z)\times y\|=\|x-z\|.\|y\|\ne0$, and therefore $(x-z)\times y\ne0$. It follows that $x\times y\ne z\times y$.

Answer 2

$$a\cdot x=0$$ is true for all $x$ perpendicular to $a$. In 2D, this is a straight line and in 3D a plane.

$$a\times x=0$$ is true for all $x$ parallel to $a$, which is a straight line in 3D.

So $x=0$ is not necessarily true.