# Given u 372 v 621 and w 238 verify the following

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12.Givenu=(−3,7,2),v=(6,2,1), andw=(−2,3,8),verify the following arithmetic propertiesof vectors.(12 marks: 4 marks each)w
The cross product is(v+w)=(53,35,43).Right side:v+w¿(3,7,2)×(6,2,1)+(3,7,2)×(2,3,8)¿(3,15,48)+(50,20,5)¿(53,35,43)The cross product isv+w=(53,35,43).Since(v+w)=(53,35,43)andv+w=(53,35,43), we can conclude that(v+w)=v+wand the distributive law holds true for cross products.
b)u∙(v+w)=u∙v+u∙wLeft side:u∙(v+w)¿(3,7,2)[(6,2,1)+(2,3,8)]¿(3,7,2)(4,5,9)¿(3) (4)+(7) (5)+(2)(9)¿12+35+18¿41Right side:u∙v+u∙w¿(3,7,2)(6,2,1)+(3,7,2)(−2,3,8)¿(3) (6)+(7) (2)+(2) (1)+(3) (2)+(7) (3)+(2)(8)¿18+14+2+6+21+16¿41Therefore,u∙(v+w)=u∙v+u∙wand the distributive law applies to the dot product of two vectors.c)(u+v)(u+v)=|u|2+|v|2+2(u∙v)Left side:(u+v)(u+v)¿[(3,7,2)+(6,2,1)][(3,7,2)+(6,2,1)]¿(3,9,3)(3,9,3)
¿(3) (3)+(9) (9)+(3)(3)¿9+81+9¿99Right side:|u|2+|v|2+2(u∙v)¿((3)2+(7)2+(2)2)2+((6)2+(2)2+(1)2)2+2(3,7,2)(6,2,1)¿62+41+2((3) (6)+(7) (2)+(2) (1))¿103+2(−2)¿99Therefore,(u+v)(u+v)=|u|2+|v|2+2(u∙v).5/5 marks13.Describe how the dot product can be used to determine whether two vectors are perpendicular. Createa question with vectors in 3-space to illustrate this property. Be sure to solve the question as well.(5 marks)is:z2
u∙v=x1x2+y1y2+z1z2=0For example, find the dot product of the following pair of vectors.Determine whether vectorsu=(1,9,3)andv=(6,2,8)are perpendicular:Find the dot product using the Cartesian formula:p∙q=(1) (6)+(9) (2)+(3)(−8)¿6+1824¿0The dot product is 0, so the vectorsp=(1,9,3)andq=(6,2,8)are perpendicular.Task 4: Application questions7/7 marks

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