Question

If `A = <8 ,1 ,4 >`, `B = <6 ,5 ,4 >` and `C=A-B`, what is the angle between A and C?

Answer 1

`≈ 72.65^@`

Explanation:

To calculate the angle between 2 vectors `ula and ulc`

Use : `costheta = (ula . ulc )/(|ula||ulc|`

where `theta" is the angle between the vectors "`

now C = A - B = (8,1,4) - (6,5,4) = (2,-4,0)

hence `ula . ulc = (8,1,4) . (2,-4,0)`

`= (8xx2) + (1xx-4) + (4xx0) = 16 - 4 + 0 = 12`

`|ula| = sqrt(8^2+1^2+4^2) = sqrt81 = 9`

and `|ulc| = sqrt(2^2+(-4)^2+0^2) = sqrt20`

`rArrtheta = cos^-1 (12/(9xxsqrt20) )≈ 72.65^@`

Answer 2

`alpha=88,58`

Explanation:

`A= <8,1,4>" "B= <6,5,4>`
`1)"find " A-B=C`
`C_x=A_x-B_x=8-6=2`
`C_y=A_y-B_y=1-5=-4`
`C_z=A_z-B_z=4-4=0`
`C= <2,-4,0>`
`2)"find dot product of A and C"`
`A*C=A_x*C_x+A_y*C_y+A_z*C_z`
`A*C=8*2+1*(-4)+4*0=16-4+0=12`
`3)"find magnitude of A and C"`
`||A||=sqrt(8^2+1^2+4^2)=sqrt(64+1+16)=sqrt81`
`||C||=sqrt(2^2(-4)^2+0^2)=sqrt(4+16+0)=sqrt20`
`4) "find angle between A and C using the dot product formula"`
`A*C=||A||*||C||*cos alpha`
`12=sqrt81*sqrt20*cos alpha`
`cos alpha=12/sqrt(81*20)`
`alpha=88,58`