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Figure $13.4$ in Tom Apostol's Calculus vol. $1$ is used to visualize cross product, and I don't understand why for fixed two vectors $A$ and $B$, the orientation of $A \times B$ vector depends on how the axes are interpreted. For example, in the image below:

  • if the red axis is $j$ and the blue one is $i$, then $A \times B$ points upwards
  • if the red axis is $i$ and the blue one is $j$ then the $A \times B$ points downwards

enter image description here

Why is that the case?

Note: I looked at Understnading left vs. right-handed systems, but it didn't help much.

S11n
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    For a right handed system i.e $i \times j = k$, then $A \times B$ will mimic the handedness of the system, i.e if $A$ is to the right of $B$ (looking top down) then $A \times B$ will point upwards and $B \times A$ will point downwards (relative to the $A-B$ plane). For a left handed system where $j\times i = k \implies i\times j = -k$, then once again $A \times B$ will mimic the handedness of the system which means if $A$ is to the right of $B$ then $A \times B$ will point downwards (minimicing the $-k$ direction, but relative to the $A-B$ plane) and $B\times A$ would point upwards. – Sam Mar 29 '23 at 17:59
  • @Sam thanks! However, how to reconcile "For a left handed system where ×=" with what grand_chat wrote in the answer below that "by definition the cross product × equals "? – S11n Mar 31 '23 at 08:05
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    Technically, according to definition $\hat{i} \times \hat{j} = \hat{k}$ where the $\hat{}$ is used to distinguish arbitrary basis vectors $i,j,k$ from the canonical standard basis vectors $\hat{i},\hat{j}, \hat{k}$. Canonically, $\hat{i} = [1,0,0]^T,\hat{j} = [0,1,0]^T,\hat{k} = [0,0,1]^T$ which is why by definition $\hat{i} \times \hat{j} = \hat{k}$. That's because we've chosen our canonical basis vectors to form a right handed system. A general system on the other hand can be chosen to be left handed or right handed as desired – Sam Apr 01 '23 at 22:22

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It's about what the green ($k$) axis is doing. If you swap the labels $i$ and $j$, but you keep the $k$ axis unchanged, then you've re-interpreted the $k$ axis to point the other way relative to the original labeling, and you've formed a new coordinate system that's not the same as before--a point in space that used to receive a positive $k$ coordinate is now receiving a negative $k$ coordinate, and vice versa.

To see this, mentally rotate the three-axis system so that the $i$ and $j$ axes are in the plane of your page/screen, with $i$ horizontal and $j$ vertical. (Or construct a physical model.) In a right-handed coordinate system, the $k$ axis will point out of the page. Now do the same thing after relabeling $i$ as $j$, keeping $k$ unchanged. You'll see that the $k$ axis flipped from pointing out of the page (right-handed coordinates) to pointing into the page (left-handed coordinates). As a concrete application of this observation, remember that by definition the cross product $i\times j$ equals $k$. So in a right-handed coordinate system this cross product points out of the page, and in a left-handed coordinate system it points the opposite direction.

Put another way, in the picture you've posted, to preserve the original coordinate system when you swap $i$ and $j$ you also need to flip the $k$ axis to point downward.

grand_chat
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  • Thanks! By "So in a right-handed coordinate system this cross product points out of the page, and in a left-handed coordinate system it points the opposite direction.", you mean in case when we fix $i$ to be the horizontal axis, and $j$ the vertical axis? If I switch the $i$ to be vertical, and $j$ to be the horizontal axis, then the $k$ will still point out of the page/screen in a left-coordinate system. – S11n Mar 31 '23 at 08:00
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    Right, in the case that $i$ is the horizontal axis and $j$ is the vertical axis you'll get this distinction between right-handed and left-handed coordinate systems. Once you've chosen $i$ to be horizontal and $j$ to be vertical, you still have freedom to orient $k$ either out of or into the page. – grand_chat Mar 31 '23 at 16:38