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# Decomposing a Vector

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Given   $$\vec{v}=3\vec{i}+4\vec{j}$$   and   $$\vec w=-2\vec i+7\vec j$$  ,  decompose  $$\vec v$$  into two vectors   $$\vec v_1$$   and   $$\vec v_2$$  , where   $$\vec v_1$$   is parallel to   $$\vec w$$   and   $$\vec v_2$$   is orthogonal to   $$\vec w$$   .

I need help understanding this...

I do know how to "get the answer" but I don't "get" the answer.

hectictar  Apr 25, 2018
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It's been awhile since I did this...so...here goes nothing

v = (3, 4)     w  =  (-2, 7)

We want to project  v onto w

projw v  =   [( v dot w) /( length of w)^2 ] * w  =  [ 22/ 53) * (-2, 7)  =  (-44/53, 154/53)

This  is   v1

To  find v2   we have    [ v - v1] =  (3, 4) - (-44/53, 154/53)  =  (203/53, 58/33)

Here's a pic :

The "why" of this always gave me some trouble, too. If I can remember what my  Calc teacher said, if we shine a beam onto v from the "top" of the diagram perpendicular to v2, it will "project" a perfect "shadow" of v onto v2. Likewise....if we shine a beam from the "right" of the diagram onto v such that the beam is perpedicular to v1, it will "project" a perfect "shadow" of v onto v1.

Sorry, hectictar.....that's about as good of an explanation as I can supply......!!!!

I think this is used in Physics to break up components of work  (as well as some other applications, too...)

CPhill  Apr 25, 2018
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Oh okay thank you for that explanation. The picture does help.

HMMM I have been thinking about it for awhile now and I think I am starting to get it!!

hectictar  Apr 26, 2018