The angle between two lines is the angle between their direction vectors.
For the first line, the direction vector is v1=2i^+1j^+2k^, and for the second line, it is v2=6i^−3j^+2k^.
The cosine of the angle θ is given by the formula cosθ=∣v1∣∣v2∣∣v1⋅v2∣.
Calculating the dot product gives (2)(6)+(1)(−3)+(2)(2)=12−3+4=13.
The magnitude of the first vector is 22+12+22=9=3.
The magnitude of the second vector is 62+(−3)2+22=36+9+4=49=7.
Substituting these into the formula, we get cosθ=3×713=2113