the x component of A is the horizontal part of A,

A 在 x 方向即水平方向上的分矢量
is the A·cos(θA)?

是 A·cos(θA)
You must've seen a lot of examples

大家在进行力学计算的时候
of that when you did all the force calculation.

应该见过很多这样的例子了吧
And Ay is equal to the length of A times sin(θA)

Ay 等于 A 的长度乘以 sin(θA)
and likewise for B; I don't feel like writing it.

B 是一样的我就不写出来了
If you do that, then A·B will be length of A, length of B times

这样分析之后 A·B 等于 A 的长度乘以 B 的长度再乘以
You've got to go back to your good old trig

回顾一下以前学的三角函数知识便知
and it'll tell you this cos cos plus sin sin is cos(θA – θB).

括号里面的是 cos(θA – θB)
So, you find it is length of A, length of B,

结果就是 A 的长度乘 B 的长度再乘以
Often, people simply say that is A·B·cos(θ)

通常大家都写成 A·B·cos(θ)
where it's understood that θ is the angle between the two vectors.

因为大家都知道 θ 是二者的夹角