If A=<6,−7,8>, B=<9,−5,−2> and C=A−B, what is the angle between A and C?

1 Answer
Narad T.
Dec 5, 2016

The angle is =54.1º

Explanation:

Let's start by calculating

→C=→A−→B

→C=⟨6,−7,8⟩−⟨9,−5,−2⟩=⟨−3,−2,10⟩

The angle between →A and →C is given by the dot product definition.

→A.→C=∥→A∥⋅∥→C∥cosθ

Where θ is the angle between →A and →C

The dot product is

→A.→C=⟨6,−7,8⟩.⟨−3,−2,10⟩=−18+14+80=76

The modulus of →A= ∥∥⟨6,−7,8⟩∥=√36+49+64=√149

The modulus of →C= ∥∥⟨−3,−2,10⟩∥=√9+4+100=√113

So,

cosθ=→A.→C∥∥∥→A∥⋅∥→C∥∥∥=76√149⋅√113=0.59

θ=54.1º

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