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**NOTE: I know the solution, but I challenge it. Specifically, I feel that the generally accepted solution to this problem**

1. The problem statement

**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)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).

## Homework Equations

Dot Product: A⋅B = ||A|| ||B|| cosθ

## 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

AS ABOVE SO BELOW

## Homework Equations

Dot Product: A⋅B = ||A|| ||B|| cosθ

Vector Addition: B = A + (B - A)

## 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!

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