THE DOT PRODUCT &

CROSS PRODUCT

Definition of DOT PRODUCT the dot product of two vectors is a number

Basically, it is one of the way to multiply two

vectors together. Geometric Definition: finding the angle between two vectors

if a and b are two vectors,

= |a||b| cos θ Algebraic Definition: determining if the two vectors are

parallel, orthogonal, or neither

if a = <a1,a2> and b= <b1,b2>,

= a1b1 + a2b2

EXAMPLES

Given: a=<1,2> and b=<2,5>

(a) Find the dot product between the two vectors.

dot product a∙b= 1(2)+ 2(5)=12

(b) Find the angle between the two vectors using dot product.

a∙b = |a||b| cos θ

cos θ = 12/ sqrt (5) *sqrt (29)

θ = 0.083 radians

Given : a=<1,2> and b=<2,5>

Definition of Cross Product

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Right-hand Rule

If and are vectors, then is a vector.(1) Direction : perpendicular to both and .

, and are in three-dimension.

(2) Magnitude : = ‖ ‖ ‖ ‖ sin𝜃

i, j, k are standard unit of vectors j x k= k x i = i x j = k x j = i x k = j x i = i x i = j x j = k x k =

i

j

k-i

-j

-k0

i

j

k

‖ a x b‖ = ‖a‖ ‖b‖ sin𝜃

Matrix Notation

=(a2b3 - + (a1b3 - a3b1 ) j + (a1b2 - a2b1 ) k

= a2b3 i + a3b1 j + a1b2 k – a2b1 k – a3b2 i – a1b3 j

If = < a1 , a2 , a3> , = <b1 , b2 , b3>, and i, j, k, are standard unit of vectors, then

a x b=

Ex. 1: If =<2, 3, 1> and = <-2, 1, 4>, compute:a) a x b b) b x a

= -(11i -10j+8k)

a) a x b =

= (3)(4)i+(1)(-2)j+(2)(1)k-(-2)(3)k-(1)(1)i-(4)(2)j=12i-2j+2k+6k-i-8j=11i-10j+8k

b) b x a =

= (1)(1)i+(4)(2)j+(-2)(3)k-(2)(1)k-(3)(4)i-(1)(-2)j=i+8j-6k-2k-12i+2j = -11i+10j-8k

a x b = -(b x a)

Properties of Cross Product

 

4. For nonzero vectors and , =0 if and only if and are parallel.

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Ex.2 : Find the angle between the two vectors a = <1,2,3> and b= <-2,1,3> using cross product.

a x b =

= (2)(3)i+(3)(-2)j+(1)(1)k-(-2)(2)k-(1)(3)i-(3)(1)j=3i-9j+5k

‖ a x b‖ = ‖a‖ ‖b‖ sin𝜃‖ a x b‖ = √3^2 +(-9)^2+(5)^2 =√115, ‖a‖ =‖b‖= √14

√115 = (√14 )(√14 ) sin𝜃𝜃= 0.873 radian

Dot Product Cross Product

Product Scalar Vector

Magnitude ||a||||b|| cos θ ||a||||b||sin θ

Direction No Perpendicular to both vectors

Dimension Any 3D

a∙b = b∙a a x b = -(b x a)

SummaryCompare Dot Product and Cross Product