# Operations with a vector not starting at the origin Suppose we have the vectors $$\vec{AB} ,\vec{AC} ,\vec{AD}$$ like in the image , and all of their coordinates are relatively to the two preprinted axes , say I want $$\vec{AB} \cdot \vec{AD}$$ , is it enough to consider the dot product between their components like they are now , or I should make a traslation of the vector so that they starts at the origin ? (I have never done linear algebra so I hope to have been clear enough)

• I think that if you want $\vec {AB}.\vec {AD}$, then you must simply consider the components as they are now. (you want $\vec {AB}.\vec {AD}$) – Devansh Kamra Apr 21 at 10:46

The components of the vector $$\vec{AB}$$ as they are now and the components after translating to the origin are the same, so it doesn't matter. Note that the components of $$\vec{AB}$$ are obtained by subtracting the coordinates of $$A$$ from those of $$B$$, this difference doesn't change when $$A$$ and $$B$$ are translated together.
In your example we have approximately \begin{align*} \vec{AB} &\approx \begin{pmatrix} (-4.5)-(-1.2) \\ 2.6-5.6 \end{pmatrix} = \begin{pmatrix} -3.3 \\ -3 \end{pmatrix}, \\ \vec{AC} &\approx \begin{pmatrix} (1.5)-(-1.2) \\ 3.1-5.6 \end{pmatrix} = \begin{pmatrix} 2.7 \\ -2.5 \end{pmatrix}, \end{align*} so that $$\vec{AB}\cdot\vec{AC} \approx (-3.3)\times2.7 + (-3)\times(-2.5) = 1.41.$$
• right right I was just confused about the fact that I had to subtracte the coordinates of $A$ from those of $B$. Thank you! – Tortar Apr 21 at 10:51