Proving Cross Product Equation: A x (B + C) = A x B + A x C

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SUMMARY

The discussion centers on proving the vector equation A x (B + C) = A x B + A x C using the properties of the cross product. Participants emphasize the importance of defining vectors A, B, and C in component form, specifically as (a_1, a_2, a_3), (b_1, b_2, b_3), and (c_1, c_2, c_3). The cross product rule is highlighted, and an example with 2-dimensional vectors is provided, demonstrating the calculation of the cross product and the determination of vector parallelism.

PREREQUISITES
  • Understanding of vector notation and components
  • Familiarity with the cross product operation
  • Knowledge of trigonometric functions, specifically sine
  • Ability to perform vector calculations in two dimensions
NEXT STEPS
  • Study the properties of the cross product in three dimensions
  • Learn how to apply the distributive property to vector operations
  • Explore vector parallelism and the conditions for two vectors to be parallel
  • Practice proving vector identities using specific examples
USEFUL FOR

Students of physics and mathematics, educators teaching vector calculus, and anyone interested in understanding vector operations and their properties.

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Problem:
Prove A x (B + C) = A x C + A x B

Related equations:
Cross product

Attempts:
Not even remotely sure where to start with this, other than I need to use the cross product rule.
 
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Try defining each vector (a_1, a_2, a_3), (b_1, b_2, b_3), (c_1, c_2, c_3) and then using the definition of the cross product in each case.
 
i have to ask that using Sinθ=u x v/|u||v|, these both vectors are 2-Dimensional u=-3i-5j,
v=4i+7j, u x v=-21i+20j ,|u|=√34,|v|=√65, Sinθ=-21i+20j/(√65√34)
question is check are they parallel?please can anyone help me for the last step
 

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