Question

If `A = <7 ,-3 ,6 >`, `B = <4 ,-7 ,-5 >` and `C=A-B`, what is the angle between A and C?

Answer 1

`alpha~=27,04 ^o`

Explanation:

`"there are five steps:"`
`1. "find A-B=C"`
`2. "find dot product A.B"`
`3. "find ||A|| (magnitude of A)"`
`4. "find ||B|| (magnitude of C)"`
`5. "use " A.C=||A||*||C||*cos alpha" formula"`

`Step-1:`
`A= <7,-3,6>" "B<4,-7,-5>`
`C=A-B`
`C_x=A_x-B_x=7-4" ; "C_x=3`
`C-y=A_y-B_y=-7+3=-4" ; "C_y=-4`
`C_z=A_z-B_z=6+5=11" ; "C_z=11`
`C= <3,-4, 11>`
`"Step-2:"`
`"now let's find the dot product of A.C"`
`A.C=A_x*C_x+A_y*C_y+A_z*C_z`
`A.C=7*3+(-3)*(-4)+6.11`
`A.C=21+12+66`
`A.C=99`
`"let's find the magnitude of A and C"`
`"Step-3:"`
`||A||=sqrt(A_x^2+A_y^2+A_z^2)" "||A||=sqrt(7^2+(-3)^2+6^2)`

`||A||=sqrt(49+9+36)" "||A||=sqrt(94)`
`"Step-4:"`
`||C||=sqrt(C_x^2+C_y^2+C_z^2)" "||C||=sqrt(3^2+(-4)^2+11^2)`
`||C||=sqrt(9+16+121)" "||C||=sqrt(146)`

`"now,let's use dot product formula"`
`"Step-5:"`
`A.C=||A||*||C||*cos alpha`
`A.C=99`
`||A||=sqrt94`
`||C||=sqrt146`

`99=sqrt94 *sqrt 146 *cos alpha`
`99=sqrt(94*146)*cos alpha`
`99=111,15*cos alpha`
`cos alpha=99/(111,15)=0,8906882591`
`alpha~=27,04 ^o`