B A "no hands" rule for the cross product (requires literacy)

AI Thread Summary
The discussion emphasizes the utility of a book's cover as a tool for visualizing the direction of the cross product between two vectors. A detailed procedure is outlined, involving positioning the vectors and orienting a book to read the title in the direction of the first vector and then opening it toward the second vector. This method allows anyone, even children, to grasp the concept of the cross product without relying on traditional hand-based techniques. The conversation also touches on the familiarity of using everyday objects, like books, to understand mathematical concepts more intuitively than using physical hand gestures. Overall, the approach highlights the importance of relatable tools in learning complex ideas.
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TL;DR Summary
How to find the vector cross product without the right-hand rule with a simple method that anyone who can read can understand intuitively.
Although it is considered unwise to judge a book by its cover, a book's cover is still useful for finding the direction of the cross product ##\mathbf{A}\times \mathbf{B}## between two given vectors. Being able to read is all that is needed. Here is a detailed procedure.

Step 1. Move one vector parallel to itself so that the vectors are tail-to tail.
Step 2. Orient the pair as a unit so that the first vector in the cross product is in the plane of a table and the second vector is pointing in a direction away from the table1.
Step 3. Place a standard book flat on the table and orient it so that its spine is perpendicular to the plane defined by the two vectors and the title is read in the direction of the first vector ##\mathbf{A}## (see Figure 1 below.)
Step 4. Open the book so that the title points along the direction of the second vector ##\mathbf{B}## (see Figure 2 below)
Step 5. Read the title and author's name. The direction from one line to the next is the direction of the cross product.
Step 6. Measure the length of the vertical component of ##\mathbf{B} ## (dotted white line) and the length of the arrow representing ##\mathbf{A}##. The product of the two lengths is the magnitude of the cross product. Clearly, the cross product is zero when the book's cover after step 4 is parallel to the table.

RHR - 3.png

Thus, anyone who can read a book2, including children, can immediately grasp how to apply this process and find the direction of the cross product without the use of right hands, advancing screws, three mutually perpendicular fingers, etc.

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1 This can be done without loss of generality. Pedants, who insist that the vectors be written in terms of a coordinate system fixed in space, may find the appropriate Euler angle rotation matrix and rotate the table instead.
2 Books printed in languages that read right to left, e.g. Hebrew, Arabic, etc., should be oriented with the title of the book in the direction of the second vector and be opened with the cover matching the direction of the first vector.
 
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Why would someone be more familiar with a book than with his hand? I have a hard time to imagine a situation where you don't have your hand with you but you have a book "handy".
 
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I tried opening a book without using me hands, but couldn't read what's on the cover and now it's all soggy with saliva. ☹
 
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nasu said:
Why would someone be more familiar with a book than with his hand? I have a hard time to imagine a situation where you don't have your hand with you but you have a book "handy".
Carrying or not carrying the tool with you is not the issue. The issue is familiarity with the use of a particular tool to visualize the link between circulation in a plane and direction perpendicular to that plane. To read, one opens a book cover (circulation) and reads from top to bottom (direction). That happens more frequently in everyday life than curling the fingers of one's right hand in the direction from the first to the second vector and considering the direction of one's thumb. For that reason, it is easier to visualize the direction of the cross product rule without a book than with a right hand.
 
kuruman said:
The issue is familiarity ...
Then we should develop a rule based on smart phones.
 
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A.T. said:
Then we should develop a rule based on smart phones.
Forget the rule. How about an app?
 
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kuruman said:
Carrying or not carrying the tool with you is not the issue. The issue is familiarity with the use of a particular tool to visualize the link between circulation in a plane and direction perpendicular to that plane. To read, one opens a book cover (circulation) and reads from top to bottom (direction). That happens more frequently in everyday life than curling the fingers of one's right hand in the direction from the first to the second vector and considering the direction of one's thumb. For that reason, it is easier to visualize the direction of the cross product rule without a book than with a right hand.
Actually, the curling and pointing happens every time you hold a stick or any other long, cylindrical object in your hand. As the winter gets colder the snow shovels will soon experience a lot of curling and pointing on their handles. The trick seems to be associating this (or the book) with the vectors. I never heard complains about curling the fingers but rather about the connections between the real object, the hand, and the abstract objects, the vectors.
 
What if the book is written in hebrew?
 
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jbriggs444 said:
What if the book is written in hebrew?
Please read footnote 2 in post #1.
 
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