- #1

Opus_723

- 178

- 3

_{x}b

_{x}+ a

_{y}b

_{y}+ a

_{z}b

_{z}= ab*cosΘ. I understand how it works in two dimensions, I think, but three is still fuzzy.

This is what I came up with for two dimensions. The angle between the vectors is simply the difference between the angle between each vector and the x-axis.

ab*cosΘ

= ab*cos(Θ

_{a}-Θ

_{b})

= ab*(cosΘ

_{a}cosΘ

_{b}+ sinΘ

_{a}sinΘ

_{b})

= a*cosΘ

_{a}b*cosΘ

_{b}+ a*sinΘ

_{a}b*sinΘ

_{b}

= a

_{x}b

_{x}+ a

_{y}b

_{y}

Now I understand intuitively that the dot product rule should work in three dimensions, since you could always orient the axes so that the two vectors lie on the x-y plane, in which case the above applies. But I'd like to see an analytical proof like the one above, except including a

_{z}b

_{z}. I would feel a lot better about this if I could see that.