Example 15
Last updated at Dec. 8, 2016 by Teachoo
Transcript
Example 15 If ๐๏ทฏ = 5 ๐๏ทฏ โ ๐๏ทฏ โ 3 ๐๏ทฏ and ๐๏ทฏ = ๐๏ทฏ + 3 ๐๏ทฏ โ 5 ๐๏ทฏ , then show that the vectors ๐๏ทฏ + ๐๏ทฏ and ๐๏ทฏ โ ๐๏ทฏ are perpendicular. Two vectors ๐๏ทฏ and ๐๏ทฏ are perpendicular if their scalar product is zero, i.e. ๐๏ทฏ . ๐๏ทฏ = 0 ๐๏ทฏ = 5 ๐๏ทฏ โ ๐๏ทฏ โ 3 ๐๏ทฏ ๐๏ทฏ = ๐๏ทฏ + 3 ๐๏ทฏ โ 5 ๐๏ทฏ ( ๐๏ทฏ + ๐๏ทฏ) = (5 + 1) ๐๏ทฏ + (โ1 + 3) ๐๏ทฏ + (โ3 + (โ5)) ๐๏ทฏ = 6 ๐๏ทฏ + 2 ๐๏ทฏ + 8 ๐๏ทฏ ( ๐๏ทฏ โ ๐๏ทฏ) = (5 โ 1) ๐๏ทฏ + (โ1 โ 3) ๐๏ทฏ + (โ3 โ (โ5)) ๐๏ทฏ = 4 ๐๏ทฏ โ 4 ๐๏ทฏ + 2 ๐๏ทฏ Now, we have to show that ( ๐๏ทฏ + ๐๏ทฏ) and ( ๐๏ทฏ โ ๐๏ทฏ) are perpendicular to each other. So, we need to show ( ๐๏ทฏ + ๐๏ทฏ) . ( ๐๏ทฏ โ ๐๏ทฏ) = 0 Taking LHS ( ๐๏ทฏ + ๐๏ทฏ) . ( ๐๏ทฏ โ ๐๏ทฏ) = (6 ๐๏ทฏ + 2 ๐๏ทฏ โ 8 ๐๏ทฏ) . (4 ๐๏ทฏ โ 4 ๐๏ทฏ + 2 ๐๏ทฏ) = (6 ร 4) + (2 ร โ4) + (โ8 ร 2) = 24 + (โ8) + (-16) = 24 + (โ24) = 0 Since ( ๐๏ทฏ + ๐๏ทฏ) . ( ๐๏ทฏ โ ๐๏ทฏ) = 0 Hence, ( ๐๏ทฏ + ๐๏ทฏ) is perpendicular to ( ๐๏ทฏ โ ๐๏ทฏ)
Examples
Example 1
Example 2
Example 3
Example 4
Example 5
Example 6
Example 7
Example 8
Example 9
Example 10
Example 11
Example 12
Example 13
Example 14 Important
Example 15 You are here
Example 16 Important
Example 17
Example 18
Example 19
Example 20
Example 21 Important
Example 22
Example 23 Important
Example 24 Important
Example 25 Important
Example 26
Example 27
Example 28 Important
Example 29 Important
Example 30 Important
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.