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I'm reading Hestenes' book "New Foundations of Classical Mechanics" as an introduction to Geometric (Clifford) Algebra. Don't worry, no physics mentioned here :)

An exercise asks to solve a vector equation, $\alpha \boldsymbol x + \boldsymbol a (\boldsymbol x \cdot \boldsymbol b)=\boldsymbol c$

The solution is given in the back of the book, yet I have absolutely no idea how to tackle this problem. FYI, the solution is: $\boldsymbol x = \frac{\boldsymbol c}{\alpha} - \frac{(\boldsymbol c \cdot \boldsymbol b)\boldsymbol a}{\alpha(\alpha+\boldsymbol a\cdot \boldsymbol b)}$

I'm familiar (but not experienced) with "vector division" and most of the vector identities you can deduce from the geometric product definitions.

The next exercise is similar, and seems to imply there are tons of these types of exercises I should be able to solve. I have searched for a more introductory text to these types of things, but didn't find anything useful for the above.

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    @GiuseppeNegro thanks for the additional reference. I'll be sure to check it out!2012-12-09

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Take inner product with $\boldsymbol b$ on both sides and solve for $\boldsymbol x \cdot \boldsymbol b$. Replace $\boldsymbol x \cdot \boldsymbol b$ in the original equation by the solution just found. Solve for $\boldsymbol x$.

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    When you put it like that, it seems so simple... Thanks.2012-12-09