Diffrence between resolving vector to components and find projections

[Mohmmad Maaitah]

Homework Statement

Resolve to components / Determine magnitude of projections

Relevant Equations

Dot product

I don't get what is the difference when I am asked to re-solve components and find projections to axes other than the Y and X
I know that the parallelogram works for the first one and the dot product for the second but what's the diffrence!

 

[haruspex]

Homework Statement: Resolve to components / Determine magnitude of projections
Relevant Equations: Dot product

I don't get what is the difference when I am asked to re-solve components and find projections to axes other than the Y and X
I know that the parallelogram works for the first one and the dot product for the second but what's the diffrence!
View attachment 332081View attachment 332082

If you resolve a vector $\vec w$ into components $\vec u, \vec v$ then $\vec w=\vec u+\vec v$.
Those components will only be the projections of $\vec w$ onto $\hat u, \hat v$ if $\vec u$ and $\vec v$ are orthogonal.

Writing $\vec u=u\hat u$ etc. and $\lambda=\hat u\cdot\hat v$,
$\vec w=u\hat u+v\hat v$
$\vec w\cdot\hat u=u+v\lambda$
etc., whence
$v=\frac{\vec w\cdot\hat v-\vec w\cdot\hat u\lambda}{1-\lambda^2}$.

 

[Mohmmad Maaitah]

If you resolve a vector $\vec w$ into components $\vec u, \vec v$ then $\vec w=\vec u+\vec v$.
Those components will only be the projections of $\vec w$ onto $\hat u, \hat v$ if $\vec u$ and $\vec v$ are orthogonal.

Writing $\vec u=u\hat u$ etc. and $\lambda=\hat u\cdot\hat v$,
$\vec w=u\hat u+v\hat v$
$\vec w\cdot\hat u=u+v\lambda$
etc., whence

I still don't get the diffrence between projection and force component.

 

[haruspex]

I still don't get the diffrence between projection and force component.

The projection of one vector on another depends only on those two vectors. It is unaffected by any other vectors under consideration.
If you are resolving into components then you need a set of directions to resolve into, $\hat u_i$, and the coefficient to use in one direction depends on the whole set of directions. If you modify $\hat u_1$ then you may find the magnitude of the component in the $\hat u_2$ direction changes.

 

[Mohmmad Maaitah]

I get it thank you sir!

The projection of one vector on another depends only on those two vectors. It is unaffected by any other vectors under consideration.
If you are resolving into components then you need a set of directions to resolve into, $\hat u_i$, and the coefficient to use in one direction depends on the whole set of directions. If you modify $\hat u_1$ then you may find the magnitude of the component in the $\hat u_2$ direction changes.