Given: Plane P1 is given by the equation 3x + 4y – 5z = 60 Plane...

Question
Given:
Plane P1 is given by the equation 3x + 4y – 5z = 60
Plane P2 is given by the equation 4x + 2y + cz = 0

Part A. Determine the value of C that makes plane P2 perpendicular to plane P1 using algebra and/or calculus techniques and justifying all work.

Part B. Determine the equations of two non-perpendicular planes (both different from the planes in Part A), and show that they are not perpendicular using algebra and/or calculus techniques and justifying all work.
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Part A. Determine the value of C that makes plane P2 perpendicular to plane P1 using algebra and/or calculus techniques and justifying all work.

P1 has normal vector (3,4,-5), and P2 has normal vector (4,2,c).
The planes are orthogonal if and only if the two normal are perpendicular which means that we need their dot product to be 0: so: 3*4+4*2+-5*c=0, so 20=5*c, so c = 4.
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        • 18. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
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        7.46mstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
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        6.42mstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
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        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 26. app/Http/Controllers/HomeworkLibraryController.php:105
        • 27. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 28. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 29. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
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        Backtrace
        • 14. app/Repositories/HomeworkLibraryRepository.php:30
        • 15. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 16. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 18. app/Repositories/HomeworkLibraryRepository.php:39
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      • select * from `subjects` where `subjects`.`id` in (6)
        1.15mstwenty4_siteHomeworkLibraryRepository.php#30
        Backtrace
        • 19. app/Repositories/HomeworkLibraryRepository.php:30
        • 20. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 21. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 23. app/Repositories/HomeworkLibraryRepository.php:39
        • 24. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 29748 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'question' order by `order` asc, `id` asc
        6.77mstwenty4_siteHomeworkLibrary.php#260
        Bindings
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        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:58
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
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        Bindings
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        Backtrace
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        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
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        2.14mstwenty4_siteHomeworkLibrary.php#260
        Bindings
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        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
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        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
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        Bindings
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