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.

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sqdancefan sqdancefan · Answer 1 · 2021-12-26T00:50:43+01:00

Answer:

yes; the distributive property applies to dot products

Step-by-step explanation:

The product of (a-b) with itself is ...

(a -b)·(a -b) = a² -2ab +b²

because the commutative and distributive properties apply to multiplication.

(a -b)·(a -b) = a(a -b) -b(a -b) = a² -ab -ba +b² = a² -ab -ab +b² = a² -2ab +b²

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The same expansion works for the dot product operation, because the commutative and distributive properties hold for that operation, as well.

(u −v)•(u −v) = u•(u -v) -v•(u -v) = u•u -u•v -v•u +v•v

= u•u -u•v -u•v +v•v = u•u -(u+u)•v +v•v = u•u -2u•v +v•v

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Additional comment

The proof of any of the properties of the dot product relies on expressing the vectors in component form. The distributive property is no exception.