a. Calculate `vec(r )=vec(a)-vec(b)+vec(c )`, where `vec(a)=5hat(i)+4hat(j)-6hat(k)`,
`vec(b)= -2hat(i)+2hat(j)+3hat(k)`, and `vec(c )= 4hat(i)+3hat(j)+2hat(k)`.
b. Calculate the angle between `vec(r )` and the `z`-axis.

Answer

a.`11hat(i)+5hat(j)-7hat(k)`; b. `cos^(-1)-7/sqrt(195)`

Explanation

a. `vec(r )= vec(a)-vec(b)+vec(c )`
`=(5hat(i)+4hat(j)-6hat(k))-(-2hat(i)+2hat(j)+3hat(k))+(4hat(i)+3hat(j)+2hat(k))`
`5hat(i)+2hat(i)+4hat(i)+4hat(j)-2hat(j)+3hat(j)-6hat(k)-3hat(k)-2hat(k)`
`11hat(i)+5hat(j)-7hat(k)`
b. `cos r` or angle between `vec(r )` and z-axis
`cos r =-(-7)/(sqrt((11)^(2)+(5)^(2)+(7)^(2)))=(-7)/sqrt(195)`
or angle `=cos^(-1) (-7)/sqrt(195)`