that no matter how your axes are oriented,

无论坐标轴指向哪里
when you ask, "How long is this arrow?"

当你问 "这个矢量有多长"
you are going to get the same answer.

你都会得到相同的结果
That means (Ax)^2 + (Ay)^2 = (Ax')^2 + (Ay')^2.

也就是 (Ax)^2 + (Ay)^2 = (Ax')^2 + (Ay')^2
That's a property of this expression.

这是这个式子的一个性质
You can square this guy and add it to the square of this guy and

你可以把这两个式子平方相加
you will find, using the magic of trigonometry,

再用一下三角函数的性质
that this is true.

你会发现这是正确的
I'm going over these points, because they're very, very important.

我详细地讲了这些是因为它们非常重要
When we do relativity,

我们学习相对论的时侯
we'll be dealing with vectors in space-time

会涉及到时空矢量的问题
and we'll find that different observers disagree on

我们会发现观测者们对于观测的结果