NOTE: I know the solution, but I challenge it. Specifically, I feel that the generally accepted solution to this problem is not intuitive. I have shown the accepted solution, and below that I have shown what I feel seems more intuitive (from a geometric standpoint) 1. The problem statement find the three angles of the triangle with vertices (2 -1 1) (1 -3 -5) (3 -4 -4). 2. Relevant equations Dot Product: A⋅B = ||A|| ||B|| cosθ 3. The attempt at a solution Analysis: define the points: ptA = (2 -1 1) ptB = (1 -3 -5) ptC = (3 -4 -4) create (position) vectors wrt origin (0,0,0): A = <2-0,-1-0,1-0> = <2, -1, 1> B = <1, -3, -5> C = <3, -4, -4> define the angles: α (between vectors A&B) (alpha) β (between vectors B&C) (beta) γ (between vectors C&A) (gamma) apply the dot product of each (position) vector wrt to one another: A*B = norm(A)*norm(B)*cos(α) ∴ α = 90.00° B*C = norm(B)*norm(C)*cos(β) ∴ β = 22.49° C*A = norm(A)*norm(B)*cos(γ) ∴ γ = 67.50° Please view: https://www.physicsforums.com/forums/calculus-and-beyond-homework.156/attachments/accepted-solution-please-check-angles-jpg.77050/?temp_hash=21a6e771909902657d7fa4fa3b0d6ae7 [Broken] 1. The problem statement AS ABOVE SO BELOW 2. Relevant equations Dot Product: A⋅B = ||A|| ||B|| cosθ Vector Addition: B = A + (B - A) 3. The attempt at a solution Flowchart: define the points: create (position) vectors wrt origin (0,0,0): find displacement vectors: -start point is point A, followed by B, C, and back to A. -then employ vector addition (final minus initial) -then employ vector addition of the entire "displacement-vectored" triangle define the angles & Please view second image: apply the dot product of each (displacement) vector wrt to one another: Analysis: *skipped to step 3 of Flowchart* find displacement vectors: "path": Start at ptA, the ptB, ptC and then return to ptA displacement vectors: B = A + (B - A) => dBA = B - A => dCB = C - B => dAC = A - C triangle described by vector sum: dBA + dCB = dAC define the angles then solve: (Please see image: https://www.physicsforums.com/forums/calculus-and-beyond-homework.156/attachments/my-understanding-of-the-vectors-png.77059/?temp_hash=21a6e771909902657d7fa4fa3b0d6ae7 [Broken]) α = acosd(sum(dAC.*dCB)*inv(norm(dAC)*norm(dCB))) ∴ α = 90.00° β = acosd(sum(dBA.*dAC)*inv(norm(dBA)*norm(dAC))) ∴ β = 157.5085° γ = acosd(sum(dCB.*dBA)*inv(norm(dCB)*norm(dBA))) ∴ γ = 112.4915° Adjustments: (I am off by 180°) adjusted_alpha = 180° - abs(alpha) => adjusted_alpha = 90.00° => adjusted_beta = 22.49° => adjusted_gamma = 67.50° END Intuitive Explanation anyone? How is my geometry wrong? What intuitive (geometric) reasons justify subtracting off 180°. Also, I am ignoring the sign of the angle, hence the abs operator above..I don't feel this is an effective solution. Can you please help me? HAPPY NEW YEAR!