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Last updated at Feb. 1, 2020 by Teachoo

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Misc 1 Write down a unit vector in XY-plane, making an angle of 30ยฐ with the positive direction of x-axis.Let the unit vector be ๐ โ We know that ๐ โ = ๐ฅ๐ ฬ + y๐ ฬ + z๐ ฬ Since the vector is in X-Y plane, there is no zโ coordinate. Hence, ๐ โ = x๐ ฬ + y๐ ฬ + 0๐ ฬ ๐ โ = ๐๐ ฬ + y๐ ฬ Unit vector in direction of x axis is ๐ ฬ & in y axis is ๐ ฬ Given that ๐ โ makes an angle of 30ยฐ with x-axis So, angle between ๐ โ & ๐ ฬ is 30ยฐ Now, we know that, ๐ โ . ๐ โ = |๐ โ ||๐ โ | cos ฮธ, Putting ๐ โ = ๐ โ , ๐ โ = ๐ ฬ & ฮธ = 30ยฐ ๐ โ .๐ ฬ = |๐ โ ||๐ ฬ | cos 30ยฐ ๐ โ .๐ ฬ = 1 ร 1 ร โ3/2 ๐ โ .๐ ฬ = โ3/2 (As ๐ โ is unit vector, |๐ โ | = 1 & ๐ ฬ is a unit vector, |๐ ฬ | = 1) (๐ฅ๐ ฬ + y๐ ฬ + 0๐ ฬ). ๐ ฬ = โ3/2 (๐ฅ๐ ฬ + y๐ ฬ + 0๐ ฬ). (1๐ ฬ + 0๐ ฬ + 0๐ ฬ) = โ3/2 ๐ฅ.1 + y.0 + 0.0 = โ3/2 ๐=โ๐/๐ Similarly, ๐ โ makes an angle of 60ยฐ with y-axis So, angle between ๐ โ & ๐ ฬ is 60ยฐ Now, we know that, ๐ โ . ๐ โ = |๐ โ ||๐ โ | cos ฮธ, Putting ๐ โ = ๐ โ , ๐ โ = ๐ ฬ & ฮธ = 60ยฐ ๐ โ .๐ ฬ = |๐ โ ||๐ ฬ | cos 60ยฐ ๐ โ .๐ ฬ = 1 ร 1 ร 1/2 ๐ โ .๐ ฬ = 1/2 (๐ฅ๐ ฬ + y๐ ฬ + 0๐ ฬ). ๐ ฬ = 1/2 (๐ฅ๐ ฬ + y๐ ฬ + 0๐ ฬ). (0๐ ฬ + 1๐ ฬ + 0๐ ฬ) = 1/2 ๐ฅ.0 + y.1 + 0.0 = 1/2 y = ๐/๐ Thus, ๐ โ = ๐ฅ๐ ฬ + y๐ ฬ = โ๐/๐ ๐ ฬ + ๐/๐ ๐ ฬ

Miscellaneous

Misc 1
Important
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Misc 2

Misc 3 Important

Misc 4

Misc 5 Important

Misc 6

Misc 7 Important

Misc 8 Important

Misc 9

Misc 10 Important

Misc 11 Important

Misc 12 Important

Misc 13 Important

Misc 14 Important

Misc 15 Important

Misc 16 Important

Misc 17 Important

Misc 18 Important

Misc 19 Important

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.