Vector Projection

The vector projection of a on b is a vector whose magnitude is the scalar projection of $u$ on $v$ with the same direction as b.

The scalar projection of $u$ on $v$ is $\frac{\mathbf{u}\cdot \mathbf{v}}{\Vert \mathbf{v}{\Vert }_2}$, so the vector projection is $\frac{\mathbf{u}\cdot \mathbf{v}}{\Vert \mathbf{v}{\Vert }_2}\cdot \mathbf{v}$.

The orthogonal to the project (“rejection”) is $\mathbf{u}-{\text{proj}}_{\mathbf{v}}(\mathbf{u})$ since we have that $\mathbf{u}={\text{proj}}_{\mathbf{v}}(\mathbf{u})+{\text{rej}}_{\mathbf{v}}(\mathbf{u})$.

Above, Welling and co are talking about this second vector “rejection” projection of ${\mathbf{c}}_{\mathbf{i}}+{\sum }_j^N{\mathbf{J}}_{\mathbf{ij}}\cdot {\mathbf{x}}_{\mathbf{j}}$ onto the tangent subspace on the hypersphere at ${\mathbf{x}}_{\mathbf{i}}$

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See also:

• Vector Space Projection — from Wolfram MathWorld
• Vector projection - Wikipedia - Wikipedia is not very helpful; kind of confusing