The angle between 2 vectors veca→a and vecb→bis given by the dot product definition
veca.vecb=∥veca∥*vecb∥costheta→a.→b=∥→a∥⋅→b∥cosθ
where thetaθ is the angle between the 2 vectors
Here,
veca=〈-5,7,6〉→a=⟨−5,7,6⟩
vecb=〈0,-4,8〉→b=⟨0,−4,8⟩
The dot product is veca.vecb=〈-5,7,6〉.〈0,-4,8〉→a.→b=⟨−5,7,6⟩.⟨0,−4,8⟩
=0-28+48=20=0−28+48=20
The modulus of veca=∥〈-5,7,6〉∥=sqrt(25+49+36)=sqrt110→a=∥⟨−5,7,6⟩∥=√25+49+36=√110
The modulus of vecb=∥〈0,-4,8〉∥=sqrt(0+16+64)=sqrt80→b=∥⟨0,−4,8⟩∥=√0+16+64=√80
costheta=(veca.vecb)/(∥veca∥*∥vecb∥)=20/(sqrt110*sqrt80)cosθ=→a.→b∥∥∥→a∥⋅∥→b∥∥∥=20√110⋅√80
costheta=0.21cosθ=0.21
theta=77.7θ=77.7º
