If A=<7,−3,5>, B=<4,−7,−5> and C=A−B, what is the angle between A and C?

1 Answer
Narad T.
Jul 13, 2018

The angle is =54.6∘

Explanation:

Start by calculating

→C=→A−→B

→C=⟨7,−3,5⟩−⟨4,−7,−5⟩=⟨3,4,10⟩

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=⟨7,−3,5⟩.⟨3,4,10⟩=21−12+50=59

The modulus of →A= ∥∥⟨7,−3,5⟩∥=√49+9+25=√83

The modulus of →C= ∥∥⟨3,4,10⟩∥=√9+16+100=√125

So,

cosθ=→A.→C∥∥∥→A∥⋅∥→C∥∥∥=59√83⋅√125=0.58

θ=arccos(0.58)=54.6∘

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