General Physics 101 PHYS Dr. Zyad Ahmed Tawfik Email : zmohammed@inaya.edu.sa Website : zyadinaya.wordpress.com

Lecture No.2 Unit Vector Notation Vectors

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

Unit Vector Notation, consider 2D axes(x , y) . 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

Unit vector notation (i,j,k) x z y Consider 3D axes (x, y, z) i Define unit vectors, i, j, k 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

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

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 + B Then, C = (Ax +Bx) + (Ay +By) Then, C =( 2 + 4 ) + (1 + 7 ) = 6 + 8 Thus, Cx = 6 & Cy = 8 b- 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

Product of Vectors

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.

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

Scalar Product of Two Vectors or (Dot product)

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

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 between A and B February 18, 2011

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

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

Derivation How do we show that Start with Then But So February 18, 2011

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

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

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

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

Solution A . B = 2 x 3 + 3 x 4 – 1x (-5) = 23 so (A.B)=23 Example 5 given two vector A = 2 i + 3 j – k and B = 3 i + 4 j – 5 k Calculate the angle between A 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}

Vector Product of Two Vectors or (Cross product)

Definition of Vector Product If A & B are vectors, their Vector (Cross) Product is defined as: C is read as “A cross B” The magnitude of vector C is AB sinθ where θ is the angle between A & B

Therefore, = AB sin θ

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

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

Derivation How do we show that ? Start with Then But So February 18, 2011

Calculating Cross Products Example 1 Find: Where: Solution:

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

B-A =6.4 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

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 and magnitude 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?

8.Given A = 2i +5j – 3K and B = 6i - 2j find the B-A ? 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 = 800 Find A x B ? 7. given two vector A = 4 i + 6 j – 2k and B = 5 i + 2 j – 7 k Calculate the angle between A and B by using dot product ? 8.Given A = 2i +5j – 3K and B = 6i - 2j find the B-A ?

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