Why can't you use dot product on vectors?

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I'm taking college Linear Algebra and Calculus right now. I'm a bit confused because in Calculus we're continuing to use the dot product just as in high school but in Lin Alg the prof said that you can't use the dot product or multiply two vectors. I understand it has something to do with closure (it needs to be a function from R_N to R_N). What's the exact reason and why am I doing different things in 2 math classes ?