Solve the following vector equations simultaneously $\vec x+\vec c \times \vec y=\vec a $ and $\vec y+\vec c \times \vec x=\vec b $.

Question

Solve the following vector equations simultaneously $\vec x+\vec c \times \vec y=\vec a $ and $\vec y+\vec c \times \vec x=\vec b $.

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I tried $$\vec c \times (\vec x+\vec c \times \vec y)=\vec c \times \vec a $$ $$\vec b- \vec y +\vec c \times (\vec c\times \vec y)=\vec c \times \vec a $$ I am stuck here.

Answer 1

First do the dot product of the second equation with $\underline{c}$

$$\Rightarrow y.c=b.c$$

Now consider the cross product of the first equation with $\underline{c}$, applying the vector triple product formula:

$$\Rightarrow \underline{c}\times \underline{x}+\underline{c}(c.y)-\underline{y}(c.c)=\underline{c}(c.a)-\underline{a}(c.c)$$

Now replace $$y.c=b.c$$ and$$\underline{c}\times \underline{x}=\underline{b}-\underline{y}$$ and rearrange for $\underline{y}$ and get $$\underline{y}=\frac{1}{1+c.c}(\underline{b}-\underline{c}(c.a-b.c)+\underline{a}(c.c))$$

So you can follow the same routine to find $\underline{x}$, or just substitute the expression for $\underline{y}$ into the first equation.