Last updated at March 11, 2017 by Teachoo

Transcript

Example 28 Let 𝑎, 𝑏 and 𝑐 be three vectors such that | 𝑎|= 3, | 𝑏|= 4, | 𝑐|= 5 and each one of them being perpendicular to the sum of the other two, find 𝑎 + 𝑏 + 𝑐. Given, | 𝑎| = 3 , | 𝑏| = 4 , | 𝑐|= 5 Also, each one of them being perpendicular to the sum of the other two 𝒂 is perpendicular to 𝒃 + 𝒄 i.e. 𝑎. ( 𝑏 + 𝑐) = 0 ⇒ 𝒂. 𝒃 + 𝒂. 𝒄 = 0 𝒃 is perpendicular to 𝒂 + 𝒄 i.e. 𝑏. ( 𝑎 + 𝑐) = 0 ⇒ 𝒃. 𝒂 + 𝒃. 𝒄 = 0 𝒄 is perpendicular to 𝒂 + 𝒃 i.e. 𝑐. ( 𝑎 + 𝑏) = 0 ⇒ 𝒄. 𝒂 + 𝒄. 𝒃 = 0 Now, | 𝒂+ 𝒃+ 𝒄 |2 = ( 𝒂 + 𝒃 + 𝒄) . ( 𝒂 + 𝒃 + 𝒄) = 𝑎. 𝑎 + 𝑎 . 𝑏 + 𝑎 . 𝑐 + 𝑏 . 𝑎 + 𝑏 . 𝑏 + 𝑏 . 𝑐 + 𝑐 . 𝑎 + 𝑐 . 𝑏 + 𝑐 . 𝑐 = 𝑎. 𝑎 + ( 𝒂 . 𝒃 + 𝒂 . 𝒄) + 𝑏 . 𝑏 + ( 𝒃 . 𝒂 + 𝒃 . 𝒄) + 𝑐 . 𝑐 + ( 𝒄 . 𝒂 + 𝒄 . 𝒃) = 𝑎. 𝑎 + (0) + 𝑏 . 𝑏 + (0) + 𝑐 . 𝑐 + (0) = 𝒂. 𝒂 + 𝑏. 𝑏 + 𝑐. 𝑐 = | 𝒂|2 + | 𝑏|2 + | 𝑐|2 = 32 + 42 + 52 = 9 + 16 + 25 = 50 So, | 𝑎+ 𝑏+ 𝑐 |2 = 50 Taking square root both sides, | 𝑎 + 𝑏 + 𝑐| = 50 | 𝑎 + 𝑏 + 𝑐| = 25 x 2 | 𝑎 + 𝑏 + 𝑐| = 5 2 So, | 𝒂 + 𝒃 + 𝒄| = 𝟓 𝟐

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About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.