What Are the Basics of Calculating Cross Products in Vector Algebra?

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The discussion focuses on the basics of calculating cross products in vector algebra, with a request for foundational understanding before tackling more complex problems. The user seeks assistance with specific calculations involving vectors x, y, and z, as well as examples involving vectors a and b. There is an emphasis on the property that the cross product is anti-commutative, meaning a cross product b equals negative b cross product a. The request for help is noted to be unclear, suggesting a need for more structured questions. Overall, the conversation highlights the importance of grasping fundamental concepts in vector operations.
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I was wondering if anyone could give me information the base of cross product all the problems I have been reading about so far have been in real depth I need to understand the basics first the problems given here are the beginning, if anyone can give me any help please do so, and thank you in advance

Homework Statement



calculate for

x cross product Y =


x cross product z =


y cross product (-z)=


a cross product b = where a= -ez and b= Bza cross product b= a= x + Y and b= Y

Homework Equations



a cross product b= -b cross product a

The Attempt at a Solution


Homework Statement


Homework Equations


The Attempt at a Solution


Homework Statement


Homework Equations


The Attempt at a Solution



Code:
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Homework Statement


Homework Equations


The Attempt at a Solution

 
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The book claims the answer is that all the magnitudes are the same because "the gravitational force on the penguin is the same". I'm having trouble understanding this. I thought the buoyant force was equal to the weight of the fluid displaced. Weight depends on mass which depends on density. Therefore, due to the differing densities the buoyant force will be different in each case? Is this incorrect?

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