We know the formula for projection of a on b $$ proj_{b} (a)=\left(\frac{a\cdot b}{||b||^2}\right)b = \left(\frac{a\cdot b}{b\cdot b}\right)b$$
and its length is called component of a in the direction of b written $$comp_b(a)=||proj_b(a)||=\frac{a\cdot b}{||b||}$$
How can we establish any relationship of above formulas with the following
Projection using vector triple product?
The sum of vector projection of a $\vec{a}$ onto a nonzero $\vec{b}$ and orthogonal projection of $\vec{a}$ onto the plane orthogonal to $\vec{b}$ is equal to $\vec{a}$.
More details are here $\rightarrow$ https://en.wikipedia.org/wiki/Vector_projection#:~:text=The%20projection%20of%20a%20vector%20on%20a%20plane,parallel%20to%20the%20plane%2C%20the%20second%20is%20orthogonal.