Question

What is the angle between `<1 , 5 , 3 >` and `< 4, 9 , 4 >`?

Answer 1

`alpha=14,07 ^o`

Explanation:

`"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`