If A=<6,4,9>, B=<7,−1,5> and C=A−B, what is the angle between A and C?

1 Answer
Narad T.
May 17, 2018

The angle is =42.9∘

Explanation:

Start by calculating

→C=→A−→B

→C=⟨6,4,9⟩−⟨7,−1,5⟩=⟨−1,5,4⟩

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=⟨6,4,9⟩.⟨−1,5,4⟩=−6+20+36=50

The modulus of →A= ∥∥⟨6,4,9⟩∥=√36+16+81=√133

The modulus of →C= ∥∥⟨−1,5,4⟩∥=√1+25+9=√35

So,

cosθ=→A.→C∥∥∥→A∥⋅∥→C∥∥∥=50√133⋅√35=0.7328

θ=arccos(0.7328)=42.9∘

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