Start by calculating
vecC=vecA-vecB→C=→A−→B
vecC=〈2,1,-3〉-〈3,4,1〉=〈-1,-3,-4〉→C=⟨2,1,−3⟩−⟨3,4,1⟩=⟨−1,−3,−4⟩
The angle between vecA→A and vecC→C is given by the dot product definition.
vecA.vecC=∥vecA∥*∥vecC∥costheta→A.→C=∥→A∥⋅∥→C∥cosθ
Where thetaθ is the angle between vecA→A and vecC→C
The dot product is
vecA.vecC=〈2,1,-3〉.〈-1,-3,-4〉=-2-3+12=7→A.→C=⟨2,1,−3⟩.⟨−1,−3,−4⟩=−2−3+12=7
The modulus of vecA→A= ∥〈2,1,-3〉∥=sqrt(4+1+9)=sqrt14∥∥⟨2,1,−3⟩∥=√4+1+9=√14
The modulus of vecC→C= ∥〈-1,-3,-4〉∥=sqrt(1+9+16)=sqrt26∥∥⟨−1,−3,−4⟩∥=√1+9+16=√26
So,
costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=7/(sqrt14*sqrt26)=0.3669cosθ=→A.→C∥∥∥→A∥⋅∥→C∥∥∥=7√14⋅√26=0.3669
theta=arccos(0.3669)=68.48^@θ=arccos(0.3669)=68.48∘
