Solving a Particular Vector Equation

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There's a problem in David Hestenes' book New Foundations for Classical Mechanics that many people have a hard time with. In this short, I show the solution.

Supporters:
David Johnston
Jason Killian
jerrud
Richard Penner
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Why can you add the inner product with b in the middle of the second term instead of at the end?

yoavboaz
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I don't quite understand, like why did a + a • b go under c • b for example, like I always get lost on these kind of things, like why you could move and remove those things in that way?

randigo
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One of the reasons GA is not that popular is that noncommutative algebra _sucks_, even worse than trigonometry. If only it was the only way it trips you...

ybcnnpq
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Neat, but what is the context of this formula? What it's uses?

linuxp
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The animation for adding the inner product and solving it was a little too quick for me, i rewinded twice and needed to pause to see what was going on…

yorailevi