MHB Can the Algebraic Result of a Dot Product be Applied to Vectors?

[veronica1999]

Could someone please take a look at my attached work?

10. Given a vector u, the familiar absolute-value notation |u| is often used for its magnitude. Thus the expressions u•u and |u|^2 both mean the same thing. What exactly do they mean?11. For any two numbers a and b, the product of a−b times itself is equal to a^2−2ab+b^2. Does this familiar algebraic result hold for dot products of a vector u − v with itself? In other words, is it true that (u − v) • (u − v) = u•u−2u•v+v•v? Justify your conclusion, trying not to express vectors u and v in component form.

 

[Jameson]

I believe the difference in #10 between u*u and |u|^2 is that u*u is a dot product and |u| is the magnitude of u which is calculated similarly to the Pythagorean Theorem but allowing for more elements. Only using words I would say this shows that the dot product of vector u is equal to the square of the magnitude of vector u.

For #11 see this Wiki page and find the category heading "Proof of the geometric interpretation". This properly is fully derived and explained step by step starting with the Law of Cosines.

I hope this helps. Both of these properties are discussed in detail on many math sites, including the Wikipedia page I linked to in this thread.

 

[veronica1999]

Thank you!