Question

If `A = <4 ,-5 ,-4 >`, `B = <-9 ,1 ,-6 >` and `C=A-B`, what is the angle between A and C?

Answer 1

`0.801 " radians" or 45.67^@`

Explanation:

If `A=< color(blue)(4, -5, -4) >`
and `B = < color(green)( -9, 1, -6) >`
then
`color(white)("XXX")C= A-B= < color(blue)4-(color(green)(-9)), color(blue)(-5)-color(green)(1), color(blue)(-4)-(color(green)(-6)) >`

`color(white)("XXXX")=< color(red)(13, -6, 2) >`

`theta_(AC) = arccos ((A*C)/(||A||xx||C||))`

`A*C= color(blue)(4)xxcolor(red)(13) +color(blue)((-5))xxcolor(red)((-6))+color(blue)((-4))xxcolor(red)(2)`
`color(white)("XXX")= 76`

`||A|| = sqrt(color(blue)(4)^2+color(blue)((-5))^2+color(blue)((-4))^2)`
`color(white)("XXX")=sqrt(57)`

`||C|| =sqrt(color(red)(13)^2+color(red)((-6))^2+color(red)(2)^2)`
`color(white)("XXX")=sqrt(209)`

`theta= arccos(76/(sqrt(57)*sqrt(209))) = arccos(4/sqrt(33))`

`color(white)("X")=0.801 " radians" or 45.67^@`