Question

What is the angle between `<3 , 2 , -1 >` and `< 1, -3 , 6 >`?

Answer 1

`alpha ~=69,23`

Explanation:

`A= <3,2,-1>" "B= <1,-3,6>`
`"steps..."`
`1-" find magnitude of A"`
`2-" find magnitude of B"`
`3-" find dot product "A*B`
`4-" use : "A*B=||A||*||B||*cos alpha`

`||A||=sqrt(3^2+2^2+(-1)^2)=sqrt(9+4+1)=sqrt14`
`||B||=sqrt(1^2+(-3)^2+6^2)=sqrt(1+9+36)=sqrt46`

`A*B=A_x*B_x+A_y*B_y+A_z*B_z`
`A*B=3*1+(2*(-3))+(-1*6)`
`A*B=3-6-6`
`A*B=-9`

`A*B=||A||*||B||*cos alpha`
`-9=sqrt14*sqrt46*cos alpha`
`cos alpha=-9/sqrt(14*46)`

`cos alpha=-9/(25,38)`
`cos alpha=0,3546099291`
`alpha ~=69,23`