A vectorial calculation Hello all, Knowing $\displaystyle \overrightarrow{z}=\overrightarrow{x}+ \overrightarrow{y}$ to calculate $\displaystyle \overrightarrow{z}^n$ where the three vectors are not collinear , $\displaystyle n\in \mathbb N$ and $\displaystyle n\geq 2 $. All the best, Integrator 
what does $\large \vec{z}^n$ mean? 
What are you given? 
The title of the thread is ”vectorial calculation” Maybe n is an exponent 
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Dan 
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This is a trivial case....The scalar product $\displaystyle \overrightarrow {v}^n=\overrightarrow{v} \cdot \overrightarrow{v} \cdots \overrightarrow{v}\cdot \overrightarrow{v}$ is what would be interesting to calculate. All the best, Integrator 
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$\vec{v}\cdot \vec{v}$ is a scalar and thus cannot be further dotted with a vector. Please stop wasting everyone's time. 
A relevant product would be the geometric algebra product, which for vectors equals $$\overrightarrow{v}\overrightarrow{w} = \overrightarrow{v}\cdot \overrightarrow{w} + \overrightarrow{v}\times \overrightarrow{w},$$ the sum of cross and dot product. But to compute further and take the $n$th power of a vector, you'd need the formula for the geometric algebra product for arbitrary blades, and that's pretty ugly. 
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The problem I posted is from another forum...It is known that if $\displaystyle \overrightarrow{z}=\overrightarrow{x}+ \overrightarrow{y}$ , then $\displaystyle \overrightarrow{z} ^2=\overrightarrow{x}^2+\overrightarrow{y}^2+2 \overrightarrow{x}\overrightarrow{y}$ and so what expressions do they have $\displaystyle \overrightarrow{z} ^3$ , $\displaystyle \overrightarrow{z}^4$ , $\displaystyle \overrightarrow{z}^5$ ,......., $\displaystyle \overrightarrow{z}^{n1}$ , $\displaystyle \overrightarrow{z}^n$? Thank you very much! All the best! Integrator 
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