#"find dot product of <1,5,3> and <4,9,4> like animation above"#

#A.B=1*4+5*9+3*4#

#A.B=4+45+12#

#A.B=61#

#" now find the magnitude of A and B"#

#||A||=sqrt(1^2+5^2+3^2)=sqrt(1+25+9)=sqrt35#

#||B||=sqrt(4^2+9^2+4^2)=sqrt(16+81+16)=sqrt113#

#"Dot product is determined by:"#

#A.B=||A||*||B||*cos alpha#

#A*B=61#

#||A||=sqrt35#

#||B||=sqrt113#

#61=sqrt35*sqrt113*cos alpha#

#61=sqrt(35*113)*cos alpha#

#cos alpha=61/sqrt(35*113)#

#cos alpha=0,9699661782#

#alpha=14,07 ^o#