Question

What is the angle between `<7 , 2 , -3 >` and `< 8 , 6 , 0 >`?

Answer 1

approximately `0.528 " radians"` or `30.28^@`

Explanation:

The angle, `theta`, between two vectors `u` and `v`
is such that
`color(white)("XXX")cos(theta)=(u*v)/(abs(abs(u))*abs(abs(v)))`

For
`color(white)("XXX")u=< u_1,u_2,u_3 > = < 7,2,-3 >`
and
`color(white)("XXX")v=< v_1,v_2,v_3 > = < 8,6,0 >`

`u*v= u_1*v_1+u_2*v_2+u_3*v_3`
`color(white)("XXx")=7xx8+2xx6+(-3)xx0=68`

`abs(abs(u)) = sqrt(u_1^2+u_2^2+u_3^2)`
`color(white)("XXx")=sqrt(7^2+2^2+(-3)^2)=sqrt(62)`

`abs(abs(v))=sqrt(v_1^2+v_2^2+v_3^2)`
`color(white)("XXx")=sqrt(8^2+6^2+0^2)=sqrt(100)=10`

Therefore
`color(white)("XXX")cos(theta)= 68/(10sqrt(62))`
`color(white)("XXXXXXX")~~0.863601` (using a calculator)
and
`color(white)("XXX")theta=arccos(0.863601) ~~ 0.528426` (radians)