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## The Dot Product & Cross Product

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**Definition of DOTPRODUCTthe 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| cosθ • 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> • Find the dot product between the two vectors. • dot product a∙b= 1(2)+ 2(5)=12**Given : a=<1,2> and b=<2,5>**(b) Find the angle between the two vectors using dot product. a∙b = |a||b| cosθ cosθ = 12/ sqrt (5) *sqrt (29) θ = 0.083 radians**Definition of Cross Product**If and are vectors, then is a vector. (1) Direction : perpendicular to both and . , and are in three-dimension. (2) Magnitude : = ‖ ‖ ‖ ‖ sin𝜃 Right-hand Rule**i, j, k are standard unit of vectors**i • 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 = k j k -i -j j -k i 0 ‖ a x b‖ = ‖a‖ ‖b‖ sin𝜃**Matrix Notation**If = < a1 , a2 , a3> , = <b1 , b2 , b3>, and i, j, k, are standard unit of vectors, then =(a2b3 - + (a1b3 - a3b1 ) j + (a1b2 - a2b1 ) k a x b= = a2b3 i + a3b1 j + a1b2 k – a2b1 k – a3b2 i – a1b3 j**Ex. 1: If =<2, 3, 1> and = <-2, 1, 4>,**compute:a) a x b b) b x a 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 = -(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.**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‖ = √3^2 +(-9)^2+(5)^2 =√115, ‖a‖ =‖b‖= √14 ‖ a x b‖ = ‖a‖ ‖b‖ sin𝜃 √115 = (√14 )(√14 )sin𝜃 𝜃= 0.873 radian**Summary**Compare Dot Product and Cross Product