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If š‘Ž āƒ— ā‰  0 āƒ—, š‘Ž āƒ—. š‘ āƒ— = š‘Ž āƒ—. š‘ āƒ—, š‘Ž āƒ— Ɨ š‘ āƒ— = š‘Ž āƒ— Ɨ š‘ āƒ— , then show that š‘ āƒ— = š‘ āƒ—.

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Question 9 If š‘Ž āƒ— ā‰  0 āƒ—, š‘Ž āƒ—. š‘ āƒ— = š‘Ž āƒ—. š‘ āƒ—, š‘Ž āƒ— Ɨ š‘ āƒ— = š‘Ž āƒ— Ɨ š‘ āƒ— , then show that š‘ āƒ— = š‘ āƒ—. Given that š’‚ āƒ—. š’ƒ āƒ— = š’‚ āƒ—. š’„ āƒ— Now, š‘Ž āƒ—. š‘ āƒ— = š‘Ž āƒ—. š‘ āƒ— š‘Ž āƒ—. š‘ āƒ— āˆ’ š‘Ž āƒ—. š‘ āƒ— = 0 š’‚ āƒ—. (š’ƒ āƒ— āˆ’ š’„ āƒ—) = 0 Thus, either (š’ƒ āƒ— āˆ’ š’„ āƒ—) = šŸŽ āƒ— Or š’‚ āƒ— is perpendicular to (š’ƒ āƒ— āˆ’ š’„ āƒ—) š’‚ āƒ— Ɨ š’ƒ āƒ— = š’‚ āƒ— Ɨ š’„ āƒ— Now, š‘Ž āƒ— Ɨ š‘ āƒ— = š‘Ž āƒ— Ɨ š‘ āƒ— š‘Ž āƒ— Ɨ š‘ āƒ— āˆ’ š‘Ž āƒ— Ɨ š‘ āƒ— = šŸŽ āƒ— š’‚ āƒ— Ɨ (š’ƒ āƒ— āˆ’ š’„ āƒ—) = šŸŽ āƒ— Thus, either (š’ƒ āƒ— āˆ’ š’„ āƒ—) = šŸŽ āƒ— Or š’‚ āƒ— is parallel to (š’ƒ āƒ— āˆ’ š’„ āƒ—) Since š‘Ž āƒ— cannot both be perpendicular and parallel to (š‘ āƒ— āˆ’ š‘ āƒ—) Therefore, the only other solution is (š‘ āƒ— āˆ’ š‘ āƒ—) = 0 āƒ— š’ƒ āƒ— = š’„ āƒ— Hence proved

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.