How I wish mathematicians were normal people

As I was trying to understand my maths book, I came across the statement (below) and went on thinking and thinking 🧐.

The statement:

"Let R=Xî+Yĵ+Zk̂ be the position vector of any point in the tangent plane at P to the surface."

Now.. What point? what tangent? Plane? And what surface? These random terms came flying from...vacuum? Uh, seriously very reader unfriendly. For some reason, academicians think it is ok to write unintelligible content. 

Let me see how I would make my students understand this.

"Girls and boys, close your eyes..and imagine a surface, a 3D surface. The shape? Just imagine a ball for convenience. We choose a point on it, dot it and name it P. Now extend this point (point P) in 2D and create a tangent plane. That is, imagine P spreading its wings in two directions so as to create a mat and hanging in air, on top of this ball, with just its belly button P touching the ball. Now about this tangent plane: go left from the belly button and choose a point R on this plane. It is located at (i.e., position vectors) R=Xî+Yĵ+Zk̂."

Yes, I do see this is way too casual. But something like this DOES work. Ofcourse this way, a 300 pages maths books would fill upto 1000 pages, but better fill the pages than confusing the students and creating disinterest in them.

Now go and continue with your boring books. Meeting dispersed.

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