If A=<3,−1,−8>, B=<5,6,−9> and C=A−B, what is the angle between A and C?

1 Answer
Narad T.
Dec 24, 2017

The angle is =96.3∘

Explanation:

Let's start by calculating

→C=→A−→B

→C=⟨3,−1,−8⟩−⟨5,6,−9⟩=⟨−2,−7,1⟩

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=⟨3,−1,−8⟩.⟨−2−7,1⟩=−6+7−8=−7

The modulus of →A= ∥∥⟨3,−1,−8⟩∥=√9+1+64=√74

The modulus of →C= ∥∥⟨−2,−7,1⟩∥=√4+49+1=√54

So,

cosθ=→A.→C∥∥∥→A∥⋅∥→C∥∥∥=−7√74⋅√54=−0.11

θ=arccos(−0.11)=96.3∘

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