Is it possible to project a vector from $\mathbb{R}^2$ onto a vector from $\mathbb{R}^3$? Given my definition of a projection (projection of $\mathbf{u}$ onto $\mathbf{v}$, $\operatorname{proj}_{\mathbf{v}}\mathbf{u}=(\frac{\mathbf{u}\cdot\mathbf{v}}{|\mathbf{v}|^2})\mathbf{v}$), it does not seem it would make sense to take the dot product of a vector in $\mathbb{R}^2$ and another in $\mathbb{R}^3$. Can anyone add some clarity to this matter? Thanks.
Projection from vector in $\mathbb{R}^2$ onto a vector in $\mathbb{R}^3$?
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linear-algebra
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0You really can't... you need three values to denote a vector in 3D and two for 2D for a good reason... – 2017-02-28
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2Well, actually, there is a natural embedding $\Bbb R^2\to\Bbb R^3,\quad(a,b)\mapsto (a,b,0)$, one might implicitly use this embedding. – 2017-02-28
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0Depends what you mean by “projection.” By one definition, a projection is a linear map such that $P^2=P$, which is not possible for any map from $\mathbb R^2$ to $\mathbb R^3$. – 2017-02-28