MHB How to Express a Force as a Sum of Parallel and Perpendicular Components?

[evinda]

Hello! (Wave)

We suppose that a force that is given by the vector $2i+j$ is applied at an object that moves at the direction $i+j$.
How can we express this force as a sum of a force that has the direction of the movement and a force that is perpendicular to the direction of the movement?

 

[I like Serena]

Hello! (Wave)

We suppose that a force that is given by the vector $2i+j$ is applied at an object that moves at the direction $i+j$.
How can we express this force as a sum of a force that has the direction of the movement and a force that is perpendicular to the direction of the movement?

Hi evinda! (Smile)

We project the force vector onto the velocity vector to find the parallel component.
And onto a normal of the velocity vector to find the perpendicular component.

Such a projection is given by:
$$\boldsymbol\pi_{\mathbf v}(\mathbf F) = \frac{\mathbf F \cdot \mathbf v}{{\|\mathbf v\|}^2} \mathbf v$$
(Thinking)