My intuition says that $(a_x + a_y +a_z) \cdot (b_x + b_y + b_z) $ would expand out to be a quadratic... Why isn't this the case?
Why can the dot product of two vectors be written in the form $a_x b_x + a_y b_y + a_z b_z$ [closed]
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1$\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$– Community BotCommented Oct 19, 2023 at 9:31
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2$\begingroup$ The formula in your question is not the dot product. $\endgroup$– Zoe AllenCommented Oct 19, 2023 at 9:39
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$\begingroup$ The dot product between two vectors is this: $v\bullet w=v_xw_x+v_yw_y+v_zw_z$. $\endgroup$– SebastianoCommented Oct 19, 2023 at 9:57
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1$\begingroup$ I believe your title has a typo (it says $a_z + b_z$ where I think it ought to have said $a_zb_z$). $\endgroup$– ArthurCommented Oct 19, 2023 at 10:19
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My intuition says that would [...] be a cubic. Why isn't this the case?
I think your intuition should say it would be a quadratic. And it kind-of is:
You could write $(\vec{a_x} + \vec{a_y} + \vec{a_z}) \cdot (\vec{b_x} + \vec{b_y} + \vec{b_z})=\vec{a_x}\cdot\vec{b_x}+\vec{a_x}\cdot\vec{b_y}+\vec{a_x}\cdot\vec{b_z}+\vec{a_y}\cdot\vec{b_x}+\vec{a_y}\cdot\vec{b_y}+\vec{a_y}\cdot\vec{b_z}+\vec{a_z}\cdot\vec{b_x}+\vec{a_z}\cdot\vec{b_y}+\vec{a_z}\cdot\vec{b_z}$ and then it indeed is quadratic.
But do note: The $\cdot$ product is not an ordinary product, and the "numbers" $\vec{a_x},\vec{a_y},\vec{a_z},\vec{b_x},\vec{b_y},\vec{b_z}$ are vectors, not numbers.
The product $\cdot$ has two important rules: if two vectors $\vec{v},\vec{w}$ are perpendicular, their dot product is $0$. If two vectors are parallel, the dot product is the (signed) product of their lengths.
And $\vec{a_x}$ is perpendicular to $\vec{b_y}$ and $\vec{b_z}$. Similarly $\vec{a_y}$ is perpendicular to $\vec{b_x}$ and $\vec{b_z}$, and $\vec{a_z}$ is perpendicular to $\vec{b_x}$ and $\vec{b_y}$, so indeed we find that $$(\vec{a_x} + \vec{a_y} +\vec{a_z}) \cdot (\vec{b_x} + \vec{b_y} + \vec{b_z})=\vec{a_x}\cdot\vec{b_x}+\vec{a_y}\cdot\vec{b_y}+\vec{a_z}\cdot\vec{b_z}=a_xb_x+a_yb_y+a_zb_z$$
Remark: I wrote $\vec{a_x}$ here for the vector with length $a_x$ in the $x$ direction. Simliarly for the other vectors. You could also write this as $a_x\vec{e_x}$ as a product of the number (not vector) $a_x$ with the vector $\vec{e_x}$ of length 1 in the $x$ direction. A number that is not a vector but just a normal number is called a scalar.
