Question 1163684
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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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