Let's start by calculating
vecC=vecA-vecB→C=→A−→B
vecC=〈4,-5,-4〉-〈5,1,-6〉=〈-1,-6,2〉→C=⟨4,−5,−4⟩−⟨5,1,−6⟩=⟨−1,−6,2⟩
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=〈4,-5,-4〉.〈-1,-6,2〉=-4+30-8=18→A.→C=⟨4,−5,−4⟩.⟨−1,−6,2⟩=−4+30−8=18
The modulus of vecA→A= ∥〈4,-5,-4〉∥=sqrt(16+25+16)=sqrt57∥∥⟨4,−5,−4⟩∥=√16+25+16=√57
The modulus of vecC→C= ∥〈-1,-6,2〉∥=sqrt(1+36+4)=sqrt41∥∥⟨−1,−6,2⟩∥=√1+36+4=√41
So,
costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=18/(sqrt57*sqrt41)=0.37cosθ=→A.→C∥∥∥→A∥⋅∥→C∥∥∥=18√57⋅√41=0.37
theta=arccos(0.37)=68.1^@θ=arccos(0.37)=68.1∘
