Applications of Dot Product and Cross Product

Overview of Dot Product and its equations: Dot Product Formula: uv = u i v i + U2V2 , where u = <ui,U2> and v= <vi,v2>

OR Equivalent Formula: uv = |u| |v| cos 9

Applications of Dot Product: The first equation is very useful in determining if the two vectors are parallel, orthogonal, or neither. The equivalent formula for Dot Product is useful in finding the angle between two vectors with an algebraic manipulation: cos 0 = (u-v) / (|u| |v|).

Quick Practice Review with Dot Products: Ex 1 u = <4,6> v = <-2,9> (a) Find the dot product between the two vectors. (b)Find the angle between the two vectors. /• w u J 6 \ > r~ ^--

Some General Applications: • Computer Programing when finding an angle. • Physics

Physics Applications of Dot Product: Physics: Work (W = F-d cos 0) and Magnetic Flux (4> = BA cos 0)

EX2 ^ "W- (yJt>) A man pushes a box diagonally with a force of 400N at an angle of 45 degrees with respect to the floor. The distance he covers is 15 meters. Calculate the amount of work d done. ^

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iv M Ex 3 A magnetic field 6 T goes through an area, with dimensions 5 m and 2 m, at an angle of 18 degrees. Find the magnetic flux through the area.

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CROSS PRODUCT Quick comparison

Cross Product - Defined only for 3D - Product = vector

- Orthogonal to both the vectors

- Vector x Vector

Dot Product - Defined for any

dimension - Product = scalar

Let d — <ax, a2, a3 > and b = < blt b2, b3 >, cross product is given by a x b = < a2b3 - a3b2, a3bx - axb3i axb2 - a2bx >

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ax a2\ $

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2 ways to derive this: T j k

axb = a, a- a. bx b2 b.

- t -Ia2 a 3 | a x b — i \u u \ \b2 b3\ J\b, ftj + k br b:

rt is a unij vector _L to both a and b y g\\, Direction - determined by right-hand rule

Geometric Interpretation for Cross Product: Let 6 = angle between a and b and assume 0 < 0 < n, then

Physics also employs some use of cross product such as IJight Hancfkule and Torque. Torque: T^sF • 1 sin 0

Right Hand Rule:^N^qvB

Some Examples: A force of 215 N at an angle o sM degreesas used to push a door .2 m from the hinge. Calculate the amount of torque generate/by the force.

An electron (charge of 1.9 xYO'19 C) goes through a magnetic field of 15 T with a velocity of 5 X 103 m/s. Calculate the force the electron experiences.

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Another way to find cross products is to use matrices (2x2) or (3x3).

Problems of Cross Products with Matrices:

1) If a = < 1,2, - 1 > and b = < 4, - 3 ,1 >, compute each of the following: a) dxb £

e ^

= - t - 5 / - H A :

+ fc(l)(-3) r / ( l ) ( l ) - t ( - l ) ( - 3 ) - fc(2)(4) = 2i-4j-3k —f— 3f— 8fe.

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-II - l | , r | l 2 1 "'U l l + * l 4 -3l = T(2 - 3) - ; ( 1 4- 4) + k(-3 - 8)

b) h a = t j k i J 4 - 3 1 4 - 3 1 2 - 1 1 2

f ( -3) ( - l ) +;(1)(1) 4- K 4 ) ( 2 ) - / ( 4 ) ( - l ) - r ( l ) (2 ) - S(-3)(l) = 3r + /+8fc + 4/ -2 f+3£

= T+5/ + llfc

OR

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