SOLUTION: For any two numbers a and b, the product of a−b times itself is equal to a2 −2ab+b2. Does this familiar algebraic result hold for dot products of a vector u − v with itself?

Question 1163684: For any two numbers a and b, the product of a−b times itself is equal to a2 −2ab+b2. Does this familiar algebraic result hold for dot products of a vector u − v with itself? In otherwords,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.
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!

Let w = u-v
The symbol • will be used for the dot product even though it looks like a multiplication symbol

(u-v) • (u-v) = w • (u-v) after replacing the first (u-v) with w.

The dot product is similar to multiplication, in that we can distribute over addition like so
w • (u-v) = w • u - w • v

note: subtraction is effectively addition with negative numbers, eg: 6 - 7 = 6 + (-7). The same applies with vectors as well

From here, we plug in w = u-v and apply another round of distribution
w • u - w • v
(u-v) • u - (u-v) • v
u • u - v • u - u • v + v • v
u • u - u • v - u • v + v • v
u • u - 2u • v + v • v

Ultimately, this shows that (u-v) • (u-v) expands out to u • u - 2u • v + v • v

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