General Physics 101 PHYS

1. General Physics101 PHYS Dr. Zyad Ahmed Tawfik Email : zmohammed@inaya.edu.sa Website : zyadinaya.wordpress.com

2. Lecture No.2 Unit Vector Notation Vectors

3. Unit Vector Notation, • A unit vector is a vector that has a magnitude of one unit and can have any direction. • Traditionally i^ (read “i hat”) is the unit vector in the x direction • and j^ (read “j hat”) is the unit vector in the y direction. |i^|=1 and |j^|=1, this in two dimensions , • and motion in three dimensions with ˆk (“k hat”) as the unit vector in the z direction

4. Unit Vector Notation, consider 2D axes(x , y) . J = vector of magnitude in the “y” direction i = vector of magnitude in the “x” direction The hypotenuse is VECTOR SUM 3j Vertical Component =3j 4i Horizontal Component = 4i

5. y j i x k z Unit vector notation (i,j,k) Consider 3D axes (x, y, z) Define unit vectors, i, j, k Examples of Use: 40 m, E = 40 i 40 m, W = -40 i 30 m, N = 30 j 30 m, S = -30 j 20 m, out = 20 k 20 m, in = -20 k

6. Important Rule If A = Ax + Ayand B = Bx + By Then, C = A + B Or, C = (Ax + Bx) + (Ay + By) (1)

7. Example, If A = 2 + & B = 4 + 7 a- Find component C ( C = A + B) b- Find the magnitude of C and its angle with the x-axis.Solution , a-We know C = A + BThen, C = (Ax +Bx) + (Ay +By)Then, C =( 2 + 4 ) + (1 + 7 ) = 6 + 8 Thus, Cx = 6 & Cy = 8b-From the Pythagorean theorem, C2 = Cx2 + Cy2 C2 = 62+ 82 = 100 C = 10. Tan θ = Cy/Cx = 8/6 = 1.333, so we find θ = 53.1 degree

8. Product of Vectors

9. There are two kinds of vector product : • The first one is called scalar product or dot product because the result of the product is a scalar quantity.  • The second is called vector product or cross product because the result is a vector perpendicular to the plane of the two vectors.

10. Why Scalar Product? – Because the result is a scalar (just a number) • Why a Dot Product? – Because we use the notation A.B

11. Scalar Product of Two Vectorsor(Dot product)

12. Scalar Product of Two Vectors is “Product of their magnitudes”.

13. Scalar Product of Two Vectors • The scalar product of two vectors is written as • It is also called the dot product • q is the angle betweenA and B

14. Scalar Product • Applying this corollary to the unit vectors means that the dot product of any unit vector with itself is one. In addition, since a vector has no projection perpendicular to itself, the dot product of any unit vector with any other is zero. î · î = ĵ · ĵ = k̂ · k̂ = (1)(1)(cos 0°) = 1 î · ĵ = ĵ · k̂ = k̂ · î = (1)(1)(cos 90°) = 0

15. Case 1, (No angle θ) • If A & B are two vectors, where A = Axi + Ayj + Azk& B= Bxi + Byj + Bzk • Then, their Scalar Productis defined as: AB = AxBx+ AyBy+ AzBz

16. Derivation • How do we show that • Start with • Then • But • So

17. Example 1, without angle θ • Given A = 3i + 2j and B = 5j – 6k • Find AB • Result is • Since, AB = AxBx+ AyBy+ AzBz • Then, AB = 3 x 0 + 2 x 5 + 0 x -6 = 0 + 10 + 0 = 10

18. Case 2, (With angle θ) • If A & B are two vectors, and θ is the angle between them, • Then, their Scalar Productis defined as: AB = AB cos θ

19. How can you calculate the angle between tow vector A and B if A = axi + ayj + azk, B = bxi + byj + bzk by using dot product ? Answer 1- first calculate dot product A . B = ax bx |ay by| az bz 2- calculate the magnitude A and the magnitude B Where magnitude 3- using equation AB = AB cos θ to find the angel θ between vector A and vector B by

20. Example 4, with angle θ • Given A = 7, θA = 600 and B = 2, θB = 800 • Find AB • Result is • Since, AB = AB cosθ • Then, AB = 7 x 2 cos 20 = 14 cos 20 = 14 x 0.94 = 13.2 , NE

21. Example 5 • given two vector A = 2 i + 3 j – k and B = 3 i + 4 j – 5 k • Calculate the angle betweenA and B by using dot product ? Solution • A . B = 2 x 3 + 3 x 4 – 1x (-5) = 23 so (A.B)=23 • Magnitude =3.74 so ( A=3.74) • Magnitude = 7.07 so (B=7.07) • Form this the(A.B)=23 and(AB=3.74X7.07=26.44) • By using equation • So { θ=29.56}

22. Vector Product of Two Vectorsor (Cross product)

23. Definition of Vector Product • If A & B are vectors, their Vector (Cross) Productis defined as: • C is read as “AcrossB” • The magnitude of vectorC is AB sinθwhereθ is the angle between A & B

24. Therefore, = AB sin θ

25. Example • Given A = 3, θA = 300 and B = 6, θB = 700 • Find • Result is • Since, A x B = AB sin θ • Then, A x B = 3 x 6 sin 40 = 18 sin 40 = 18 x 0.643 = 11.6 NE

26. Applying this corollary to the unit vectors means that the cross product of any unit vector with itself is zero. • î × î = ĵ × ĵ = k̂ × k̂ = (1)(1)(sin 0°) = 0

29. Derivation • How do we show that ? • Start with • Then • But • So

30. Calculating Cross Products Example 1 Find: Where: Solution:

31. Calculating Cross Products Calculate torque given a force and its location Solution:

32. Given A = 2i +5j – 3K and B = 6i + 2j+K find the B-A ? Answer B-A= (6i + 2j+K ) – (2i +5j – 3K) B-A= 6i + 2j+K - 2i -5j +3K B-A= 4i - 3j+4K B-A = = = B-A =6.4

33. Questions • 1. what does it mean: 1- Scalar Product 2- Cross Product • 2. given two vector A = 3i + 2j – K and B = 2i + 5j –3k find A.B? • 3. The magnitude of A = 3, θA = 300 andmagnitude of B = 6, θB = 700 Find is a) A x B & b) A + B • 4. Two vector A = 3i -6j – 5K and B = 2i + 3j –2k find the magnitude A+B?

34. 5. Two vector A = 5i -7j +10K and • B = 2i + 3j –2k find • a)- A x B b)- A .B • 6.Given A = 9, θA = 400 and B = 5, θB = 800Find A x B ? • 7. given two vector A = 4 i + 6 j – 2k and B = 5 i + 2 j – 7 k • Calculate the angle betweenA and B by using dot product ? • 8.Given A = 2i +5j – 3K and B = 6i - 2j find the B-A ?

35. Thank You for your Attention