html5-img
1 / 11

The Dot Product & Cross Product

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

Download Presentation

The Dot Product & Cross Product

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. The Dot Product &Cross Product

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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)

  9. Properties of Cross Product 4. For nonzero vectors and , =0 if and only if and are parallel.

  10. 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

  11. Summary Compare Dot Product and Cross Product

More Related