Question

If `A = <6 ,4 ,-3 >`, `B = <7 ,1 ,-4 >` and `C=A-B`, what is the angle between A and C?

Answer 1

The angle between A and C is `cos^-1(3/sqrt(671))`.

Explanation:

Let us consider A=(6,4,-3) into a vector `vec A=6hati+4hatj-3hatk` and similarly with B as `vecB=7hati+hatj-4hatk`.
For C=A-B as given above for this,
The difference of the vectors `vecA` and `vecB` is given by
`vecA-vecB=(a_1-b_1)hati+(a_2-b_2)hatj+(a_3-b_3)hatk`

Substituting the values and calculating we get,
`:.C=-hati+3hatj+hatk`
So, now we can find the angle between the A and C and this is given by,
`theta=cos^-1{{a_1b_1+a_2b_2+a_3b_3}/{sqrt((a_1)^2+(a_2)^2+(a_3)^2)sqrt((b_1)^2+(b_2)^2+(b_3)^2)]}`

substituting and calculating we get,

the angle between the A and C is `theta = cos^-1(3/sqrt(671))`.