Question

If `A = <3 ,2 ,5 >`, `B = <5 ,-7 ,8 >` and `C=A-B`, what is the angle between A and C?

Answer 1

`theta=87.12`

Explanation:

`"Strategy...."`

`1 diamond" find " C=A-B`
`2 diamond" find || A|| (magnitude of A)"`
`3 diamond" find ||C|| (magnitude of C)"`
`4 diamond" find A.C (dot product of A and C)"`
`5 diamond" use A.C=||A||.||C||. cos" theta " (dot product formula)"`

`1diamond" ......................................................"`

`A_x=3" ; "A_y=2" ; "A_z=5`

`B_x=5" ; "B_5=-7" ; "B_z=8`

`C= < (A_x-B_x),(A_y-B_y),(A_z-B_z) >`

`C= < (3-5),(2+7),(5-8)`

`"we obtain :"`
`C= < -2,9,-3>`

`2diamond" ........................................................"`

`||A||=sqrt(A_x^2+A_y^2+A_z^2)=sqrt(3^2+2^2+5^2)=sqrt38`

`3diamond" .........................................................."`

`||C||=sqrt(c_x^2+C_y^2+C_z^2)=sqrt((-2)^2+9^2+(-3)^2)`

`||C||=sqrt(4+81+9)=sqrt94`

`4diamond" ..........................................................."`

`A*C=A_x*C_x+A_y*C_y+A_z*C_z`

`A*C=3*(-2)+2*9+5*(-3)=-6+18-15`

`A*C=-3`

`5diamond" ............................................................"`

`-3=sqrt38*sqrt94*cos theta`

`cos theta=-3/(sqrt(38*94))`

`cos theta=-3/(59.766)`

`cos theta=0.0501957635`

`theta=87.12`